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THE   PRINCIPLES 


THEEMODYl^AMICS, 


SPECIAL  APPLICATIONS  TO  HOT-AIR,  (IAS 
AND  STEAM  ENGINES. 


BY 

ROBERT    RONTGEN. 

Teachek  in  the  Polytechnic  School  at  Kemscheid. 


TRANSLATED,  REVISED,  AND  ENLARGED 
BY 

A.  JAY  DU  BOIS,  PH.D., 

Peofessoe  of  Dtnamio  Enqineeeincj  in  the  Sheffield  Scientific  School  of  Tale  College. 


7i\r  TWO  PARTS. 


PART  I.     General  Principles— Hot- Air  and  Gas  Engines. 
PART  II.   Heat,  Steam  and  Steam  Engines. 


WITH  103  WOOD-CUTS  m  THE  TEXT. 


NEW   YORK: 

JOHN    WILEY    &    SON 
15    As TOR    Place. 


(Q  U)  ^  ^  --  I 


Bt  JOHN  WILEY  &  SONS. 


/Z 


; 


PREFACE. 

In  the  presentation  of  tlie  jDresent  work  to  tlie  Engineering 
Profession  and  to  our  teclinical  Scliools,  a  few  words  of  intro- 
duction seem  necessary. 

There  are  several  excellent  works  upon  the  subject  of  Ther- 
modynamics, in  English,  but  none  with  which  the  writer  is  ac- 
quainted, sufficiently  wide  in  its  scope  and  practical  in  its 
applications,  and  at  the  same  time  adapted  in  its  mode  of 
treatment  to  the  needs  of  heginners.  The  subject  is  thus  one  of 
the  most  difficult  for  the  student  to  get  hold  of  in  the  scheme 
of  our  engineering  schools,  and  the  effort  to  teach  it,  so  far  as 
the  writer's  experience  goes,  is  seldom  productive  of  satisfac- 
tory results. 

It  is  to  meet  this  want  that  the  present  work  is  offered  to  the 
public.  The  writer  has  used  the  work  of  Eontgen  in  his  classes 
for  several  years,  and  with  good  success.  The  treatment  is  full 
and  practical  and  the  presentation  such  as  to  offer  but  little 
difficulty  to  an  earnest  reader.  The  notation  employed  is  that 
used  by  Zeunee,  so  that  the  book  forms  a  good  introduction  to 
the  " Wdrme-tlieorie" 

During  these  years  the  work  of  Kontgen  has  grown  upon  the 
writer's  hands  into  its  present  proportions,  and  it  becomes 
proper  here  to  point  out  at  least  those  portions  for  which  the 
German  author  is  not  responsible.  In  general  the  work  of 
Eontgen  is  comprised  by  the  large  print  only,  while  all  the  rest 
is  from  other  sources. 

Of  these,  apart  from  the  writer's  own  additions,  the  most 
noteworthy  are  the  two  lectures  by  Peoe.  Yeedet,  which  have 
been  introduced  as  an  introduction  to  the  work.     They  form, 


iv  PREFACE. 

with  the  Notes  and  Additions,  an  admirable  summary  of  the 
whole  field,  and  being  in  a  popular  form,  will,  it  is  hoped, 
awaken  an  interest  in  one  of  the  most  important  of  the  more 
recent  developments  of  physical  science. 

In  Chapter  XIII.  we  have  given  a  very  excellent  abstract  of 
MoNS.  Peknolet's  work  "L'Air  Comprime,"  for  which  we  are 
indebted  to  Mr.  Bailey  Willis,  M.  E.  It  will  be  found  of  great 
interest,  and  the  diagram  given  at  the  end  of  the  chapter  will 
be  found  of  great  practical  value. 

In  the  Appendix  to  Chapter  XXIII.,  we  have  given  Zeunee's 
theory  of  siqjerMated  steam — by  far  the  best  and  latest  work  upon 
this  very  important  subject. 

As  to  other  additions,  we  have  added  here  and  there  to  the 
text  of  our  author,  matter  which  seemed  desirable,  distinguish- 
ing all  such  additions  by  fine  print  and  brackets  ;  have  appended 
"  questions  for  examination  "  to  many  chapters,  as  well  as  added 
many  selected  "  examples  for  practice,"  reduction  tables,  etc., 
all  of  which  are  calculated  to  aid  the  teacher  and  student. 
The  steam  tables  at  the  end  of  the  work  are  taken  from  Zeu- 
ner's  "  Wdrme-theo7ne." 

As  regards  the  extent  of  the  work,  it  will  be  found  consider- 
ably more  than  can  be  read  completely  by  any  class.  This  need 
cause  no  trouble  to  the  instructor.  The  principles  are  completely 
laid  down  in  the  first  six  chapters  of  the  first  and  second  parts. 
The  rest,  consisting  merely  of  the  applications  of  these  princi- 
ples, can  be  pursued  at  such  length  as  may  seem  proper  in  any 
case.  We  consider  it  a  positive  advantage  to  the  student,  who 
is  expected  to  make  use  of  the  principles  he  acquires,  to  have  a 
text  book  so  full  and  comprehensive  that  it  shall  serve  as  a 
book  of  reference  as  well,  and  point  out  the  method  to  be  pur- 
sued in  the  investigation  of  any  problems  which  may  occur  in 
future  practice. 

For  several  reasons  it  has  not  been  thought  well  to  convert 
the  French  measures  into  English.  Those  who  wish  to  become 
familiar  with  the  literature  of  the  subject,  must  be  able  to  use 


PREFACE.  Y 

the  French  system  easily.  No  graduate  of  our  Technical 
Schools  should  be  without  a  thorough  knowledge  of  it.  For 
the  practical  use  of  the  formulae  and  results  in  daily  work,  the 
reduction  tables  we  have  given  will  be  found  to  answer  every 
requirement. 

The  method  of  the  author  requires  only  a  knowledge  of  al- 
gebra and  no  use  is  made  of  the  calculus.  This  fact  will  per- 
haps gain  for  the  work  readers  who  have  long  desired  to  obtain 
some  insight  into  the  subject,  but  have  been  unable  to  read  the 
works  hitherto  published  upon  it. 

The  effort  throughout  has  been  to  aid  both  teacher  and  stu- 
dent in  their  work,  and  to  imjDart  such  a  knowledge  of  the  sub- 
ject as  shall  render  it  practically  serviceable. 

Sheffiel    Scientific  School  of  Yale  College, 
June  11th,  1880. 


G-ENEEAL    CONTENTS. 


INTEODUCTION. 

TWO  LECTURES  BY  PEOF.  E.  VERDET  UPON  THE  MECHANICAL  THEORY  OF  HEAT. 

PAGE 

First  Lecture 3 

Second  Lecture 29 

Notes  and  Additions  to  the  Lectures. 59 


PAET  I.— GENEKAL  PEINCIPLES— HOT-AIR  AND 

GAS  ENGINES. 

CHAPTER  I. 

Generation  of  heat  by  meclianical  work,  and  the  reverse.     Determination  of 
the  mechanical  equiyalent  of  heat  by  exijeriment 101 

CHAPTER  II. 

Heat  a  kind  of  motion. Ill 

CHAPTER  III. 

Inner  and  outer  work.    Latent  and  specific  heat 121 

CHAPTER  IV. 

Expansion  of  gases.    Specific  heat  of  gases.     Determination  of  mechanical 

equivalent  of  heat 137 

vii 


viii  GENERAL  CONTENTS. 

CHAPTEE    V. 

PAGE 

Heat  curves  and  the  mechanical  work  which  a  gas  performs  during  expan- 
sion and  receives  during  compression 154 


CHAPTER  yi. 

The  simple  reversible  cycle  process.     Illustration  of  the  process  by  anal- 
ogous principles  of  mechanics 1'^ 


CHAPTER  VII. 

General  law  of  the  relation  between  pressure  and  volume  of  a  gas.     Graphi- 
cal representation  of  the  inner  work 


CHAPTER  VIII. 

Comparison  of  the  hot-air  engine  and  steam  engines.     Various  kinds  of  hot- 
air  engines 


CHAPTER  IX. 

Theory  of  those  open  and  closed  hot-air  engines,  in  which,  during  each 
period,  the  air  goes  through  a  simple  reversible  cycle  process 228 


CHAPTER  X. 

The  hot-air  engines  of  Laubereau  and  Lehmann 256 

CHAPTER  XI. 

Gas  engines,  especially  those  of  Otto  and  Langen 380 

CHAPTER  XII. 

Formulae  for  the  velocity  with  which  air  flows  out  of  vessels 313 

CHAPTER  XIII. 

Air  compressors  and  compressed  air  engines 322 

Examples  for  practice > 347 

Reduction  tables 352 


GENERAL   CONTENTS.  ix 

PAET  II.— HEAT,  STEAM,  AND  THE  STEAM  ENGINE. 
CHAPTER  Xiy. 

PAGE 

The  action  of  heat  in  eyaporation.     General  properties  of  steam.     Pressure 
of  saturated  steam 359 


CHAPTER  XY. 

Heat  of  the  liquid.     Total  heat.     Inner  and  outer  heat  of  vaporization. 

Heat  of  the  steam 373 


CHAPTER  XVI. 

Calculation  of  specific  steam  volume.     Empirical  formula  for  the  inner  and 
outer  latent  heat,  as  also  for  the  density  of  steam 385 


CHAPTER  XVII. 

Curve  of  constant  steam  weight.    Empirical  formulae.    Deportment  of  steam 
when  it  expands  performing  work , 


CHAPTER  XVIII. 

Heat  curves  of  steam  and  liquid  mixtures.     Construction  of  the  same. 

Technical  applications 415 

Appendix  to  Chap.  XVIII 439 


CHAPTER  XIX. 

Other  changes  of    condition  of   steam  and  liquid  mixtures  of  practical 
importance 448 

CHAPTER  XX. 

Theory  of  the  condenser 460 

CHAPTER  XXI. 

'J'he  flow  of  steam  and  hot  water  through  orifices 470 

CHAPTER  XXIL 

Constructions  which  depend  upon  similar  principles.     The  injector . 489 


X  O  ENEMA  L   CONTENTS. 

CHAPTER  XXIII. 

PAGE 

Superheated  steam 506 

Appendix  to  Chap.  XXIII. — Theory  of  superheated  steam 517 


CHAPTEE  XXIV. 

The  more  important  principles  which  should  govern  the  construction  of  the 

engine 546 


CHAPTEE  XXV. 

Complete  calculation  of  the  steam  engine 565 

Examples  for  practice 615 

Steam  tables , ,  i , , , , , 618 


CONTENTS. 


INTEODUCTION. 

PAGE 

First  lecture 3 

Second  lecture 29 

Notes  and  additions 59 


PAET    I.— GENEEAL    PEINCIPLES— HOT-AIE    AND 

GAS  ENGINES. 

CHAPTEE  I. 

Beat  generated  by  mechanical  action 101 

Approximate  detennination  of'  the  heat  generated  by  friction 101 

Experiment  of  Davy 103 

Mayer,  the  founder  of  the  theory ■. .  103 

Exact  determination  of  the  mechanical  equivalent  by  Joule 104 

Hirn's  determination 106 

Generation  of  mechanical  work  by  heat 106 

Questions  for  examination ...   110 

CHAPTER  11. 

Redtenbacher's  theory Ill 

Other  views,  as  to  the  nature  of  heat 112 

Heat  conduction  and  radiation 115 

Questions  for  examination 120 

CHAPTER  III. 

Different  works  performed  by  heat 121 

Outer  and  inner  work 123 

xi 


xfi  CONTENTS. 

PAGE 

Specific  volume — specific  pressure 123 

Fundamental  equations  of  the  mechanical  theory  of  heat 133 

Change  of  signs  of  the  terms  in  Equation  1 134 

Increased  pi'essure  raises  the  melting  point 126 

Heat  imparted  to  gases  under  different  conditions  for  equal  rise  of  tempera- 
ture   136 

Change  of  the  equations  when  heat  is  abstracted 127 

Specific  heat 138 

Volume  capacity 130 

The  disgregation  work  small  in  solids  and  liquids 133 

Specific  heat  for  constant  volume  and  pressure 134 

Questions  for  examination ^ 135 

CHAPTER  IV, 

Expansion  of  gases  when  heated 137 

Mechanical  work  of  air  during  expansion 140 

Heating  under  constant  volume 141 

Absolute  zero  of  temperature 143 

Calculation  of  the  mechanical  ec^uivalent 143 

Increase  of  expansive  force  by  compression 146 

Mariotte's  law 146 

Mariotte's  and  Gay  Lussae's  laws  combined . .  • 148 

Questions  for  examination 153 

CHAPTEE  V. 

Isothermal  curve 154 

Mechanical  work  which  the  air  performs  during  expansion,  and  receives 

during  compression 155 

Heat  imparted  or  abstracted  during  expansion  or  compression,  according  to 

Mariotte's  law 156 

Isodynamic  curve 157 

Adiabatic  curve 158 

Outer  work  performed  by  air  expanding  adiabatically 164 

Adiabatic  compression  of  air 167 

Table  for  adiabatic  change  of  air 171 

Transference  from  one  adiabatic  to  another 173 

Equations  for  expansion  under  constant  pressure 175 

Questions  for  examination = 178 


CHAPTEE  VI. 

The  simple  reversible  cycle  process 179 

Intermediate  body 181 

Transformation  of  Equation  XXVIII 183 

Illustration  by  mechanical  principles 185 

Questions  for  examination 188 


PHYSICS  DEFT. 

CONTENTS.  xiii 

CHAPTEE  VII. 

PAGE 

Eeview  of  preceding  principles 189 

General  relation  between  volume  and  pressure 193 

Graphical  representation  of  the  inner  work 199 

Questions  for  examination 201 

Notation  of  most  frequent  use 302 

Recapitulation  of  principal  f ormulsB 203 

CHAPTER  VIII. 

Efficiency  of  the  steam  engine 206 

Work  of  one  kilogram  of  water ....  207 

Historical  note  upon  hot-air  engines 209 

Comparison  of  work  of  hot-air  and  steam  .    210 

I.  Open  hot-air  engine  with  exterior  fire 213 

Description  of  Ericsson's  engine 213 

Method  of  action 214 

Variation  of  pressure 217 

II.  Open  hot-air  engine  with  interior  fire , 223 

III.  Closed  hot-air  engine 227 

CHAPTEE  IX. 

Theory  of  open  and  closed  hot  air  engines 228 

Transformation  of  formulaj 240 

Maximum  delivery 241 

The  regenerator 247 

Absolute  maximum  delivery 248 

Formulae  for  hot-air  engines — simplest  form 251 

Construction  of  Unger's  engine ; 254 

CHAPTEE  X. 

Description  of  the  hot-air  engine  of  Laubereau 356 

Theory  of  this  engine 260 

Delivery  of  this  engine 264 

Dimensions  for  a  given  delivery 266 

The  hot-air  engine  of  Lehmann 267 

Calculation  of  the  pressures  in 272 

Delivery  of  the  engine 274 

Comparison  of  Ericsson's,  Laubereau's,  and  Lehmann's  engines 279 

CHAPTEE  XI. 

Gas  engines 280 

General  remarks 282 


xiy  CONTENTS. 

PAGE 

Delivery  and  consumption  of  gas  of  the  Lenoir  engine 286 

Consumption  of  cooling  water 290 

Description  of  the  atmospheric  gas  engine 294 


CHAPTER  XII. 

Formulae  for  the  velocity  of  efiiux  of  air  from  vessels 313 

CHAPTER  XIII. 

AIR   COMPEESSOES   AND   COMPEESSED  AIR  ENGINES. 

Work  of  compression 322 

Volume  of  the  compressing  cylinder 328 

Pinal  temperature 324 

Compression  in  two  cylinders  with  intermediate  reservoir 324 

Friction  of  air  in  pipes , 327 

The  compressed  air  engine 327 

1.  Complete  Expansion. 

Final  temperature 328 

Disposable  work 328 

Volume  of  cylinder 329 

"Weight  of  air  per  second 329 

Efficiency  of  the  compressor  and  engine 329 

Construction 331 

Two  cylinders  and  intermediate  reservoir 332 

2.  Full  Pressure. 

Disposable  work 335 

Final  temperature 336 

Comparison  of  efficiency  for  full  pressure  and  complete  expansion 336 

Weight  of  air  per  second 337 

Volume  of  cylinder ; 338 

3.  Incomplete  Expansion. 

Disposable  work 338 

Weight  of  air  per  second 339 

Volume  of  cylinder 339 

Graphic  rejjresentation  of  the  theory  of  compressed  air  engines 340 

Construction  of  the  diagram 342 

Variation  of  work  with  difEerent  degrees  of  expansion 345 

Influence  of  the  vapor  of  water  in  the  air 345 

Examples  foe  Practice 347 


CONTENTS.  XV 

REDUCTION  TABLES.   ■ 

PAGE 

I.  For  converting  meters  into  inches 353 

II.     "  "  inches     "    centimeters 353 

III.  "  "  French  measures  into  English 353 

IV.  "  "  English       "  "    French 353 

V.     "  "  kilograms  into  pounds  avoirdupois   (or   calories  into 

heat  units — Centigrade  degrees.) 353 

VI.     "  "  avoirdupois  pounds   into  kilograms  (or  heat   units — 

Centigrade  degrees — into  calories.) 354 

VII,     "  "  meter-kilograms  into  foot-pounds 354 

VIII.     "  "  foot-pounds  into  meter-kilograms 354 

IX.     "  "  kilograms  per  sq.  centimeter  into  pounds  per  sq.  inch .  355 

X.     "  "  pounds  per  sq.  inch  into  kilograms  per  sq.  centimeter.  355 

XI.     "  "  atmospheres  into  pounds  and  kilograms 355 

XII.     "  "  calories  into  Farhenheit  heat  units 356 

XIII.     "  "  Fahrenheit  heat  units  into  calories 356 


PAET  II.— HEAT,  STEAM  AND  THE  STEAM  ENGINE. 


CHAPTER  XIV. 

The  different  effects  of  heat  in  evaporation 359 

General  properties  of  steam 363 

Empirical  formulae  for  the  pressure  and  temperature  of  saturated  steam 366 

Questions  for  examination 373 


CHAPTER  XY. 

Specific  heat  and  heat  of  the  liquid 373 

Total  heat  and  heat  of  vaporization 377 

Inner  and  outer  heat  of  vaporization.     Heat  of  the  steam 379 

Questions  for  examination ., 384 


CHAPTER  XYI. 

Calculation  of  specific  steam  volume 385 

Customary  terms  and  notation  for  steam 386 

Steam  volume  calculated 388 

Calculation  of  the  outer  and  inner  latent  heat 393 

Empirical  formulae  for 393 

Density  of  saturated  steam 396 

Questions  for  examination 398 


xvi  CONTENTS. 

CHAPTER  XVII. 

PAGE 

Curve  of  constant  steam  weight 399 

Curve  of  saturation.     Critical  temperature 401 

Deportment  of  steam  when  it  expands  performing  work 402 

Heat  imparted  or  abstracted  for  great  differences  of  temperature 410 

Deportment  of  other  vapors 412 

Questions  for  examination 414 


CHAPTER  XVm. 

ISOTHERMAL   CURVE   FOR   STEAM. 

Form  of  the  curve 415 

Outer  and  inner  work  during  expansion 415 


ISODYNAMIC   CURVE. 

Equation  and  construction  of  the  curve 418 

Outer  work — heat  required 420 

ADIABATIC   CURVE. 

Equation  and  construction  of  the  curve 421 

Approximate  foi-mute  for 432 

"Work  of  steam  expanding  adiabatieally 436 

Appendix  to  Chap.  XVIII 439 

Questions  for  examination 447 

CHAPTEE  XIX. 

The  deportment  of  steam  and  liquid  mixtures  when  heat  is  imparted  or  ab- 
stracted under  constant  volume 448 

Mixture  of  steam  quantities  when  in  different  conditions 455 

Questions  for  examination 459 


CHAPTEE  XX. 

« 

Theory  of  the  surface  condenser 460 

Theory  of  the  jet  condenser 465 


CHAPTEE  XXI. 

Flow  of  steam  through  orifices 470 

Velocity  of  efflux  when  heat  is  neither  added  nor  abstracted 473 

Transformation  of  preceding  equations 475 

Another  expression  for  the  efflux 477 


CONTENTS.  XYii 

PAGE 

Steam  volume  and  weight  per  second 479 

Efflux  of  hot  water 481 

in  which  hot  water  issues  with  the  same  velocity  as  cold 485 


CHAPTER  XXII. 

The  injector 489 

Description  of , 490 

Theory  of  the  apparatus 491 


CHAPTEK  XXIII. 

Superheated  steam 506 

Illustration  by  diagram  of  saturated  and  superheated  steam 507 

The  law  of  Hii-n 508 

Calculation  of  the  specific  volume  of  superheated  steam  by  Hirn's  law 509 

Appendix  to  Chap.  XXIII 

Zeuner's  theory  of  superheated  steam 517 

Preliminary  investigation , 518 

Deduction  of  the  "  equation  of  condition  " 524 

Test  of  the  new  equation 526 

Fundamental  equations  of  the  mechanical  theory  of  heat  applied  to  super- 
heated steam  531 

Recapitulation  of  f ormuliB 535 

Applications 536 

Adiabatic  curve 536 

Isodynamic  curve 538 

Isothermal  curve  539 

Heating  under  constant  volume 542 


CHAPTEE  XXIY. 

The  more  important  principles  which  should  govern  the  construction  of  the 

steam  engine 546 

The  cycle  process  of  the  perfect  steam  engine  and  the  disposable  work 550 

The  imperfection  of  the  cycle  process 555 

a.  Condensing  engine — surface  condenser 558 

h.  Condensing  engine — jet  condenser 559 

c.  Non-condensing  engine  with  feed-pump 559 

d.  Non-condensing  engine  with  injector 560 


CHAPTER  XXY. 

Complete  calculation  of  the  steam  engine 565 

Indicated  deliA^ery 565 

B 


xviii  CONTENTS. 

PAGE 

Action  of  the  steam  in  the  cylinder 565 

1.  Travel  of  piston  up  to  beginning  of  expansion 573 

2.  "                  "                     "    compression 573 

3.  "                  "    end  of  expansion 574 

4.  "                  "    end  of  compression 574 

Steam  volume  per  stroke — degree  of  expansion  and  compression 575 

Work  of  the  driving  steam 577 

"Work  of  the  back  pressure 579 

Values  of  ^,  p.^,  and  p.^ * 581 

Cross-section  of  steam  passages 583 

Shortest  form  of  the  formula  for  indicated  deliveiy 584 

Work  of  the  engine  when  disconnected 586 

Theory  of  the  crank 586 

Calculation  of  the  weight  of  fly-wheel 589 

Dimensions  of  the  rim  and  arms 594 

Diameter  of  journals — weight  of  shaft 595 

Mean  effective  pressure  necessary  for  overcoming  the  resistance  of  friction. .  596 

Mean  efEective  pressure  required  for  working  the  cold  water  and  air  pumps..  597 

Useful  delivery 599 

Steam  weight  per  hour COO 

Quantity  of  fuel  per  hour 603 

Cost  of  a  horse  power  per  hour 603 

Calculation  of  a  projected  steam  engine 605 

Examples  for  practice. .    615 

Steam  Tables. 

I.  Expansive  force  of  steam  for  temperatures  from  —  33°  up  to  +  230°  C, 

according  to  Regnault 619-621 

II.  Principal  table  for  saturated  steam 633-627 

Ila.  Saturated  steam 638-639 

III.  Auxiliary  tables  for  saturated  steam  (Zeuner) 630 

IV.  Properties  of  saturated  steam 631-637 


INTRODUCTION. 


FIRST  LECTURE, 


CONTENTS. 


I.  Object  of  the  Lectures.— Equality  of  work  and  of  living  force.— General  conclusions : 
Equality  of  the  worli  of  the  moving  and  of  the  resisting  forces  in  machines  moving  with  constant 
velocity;  Impossibility  of  perpetual  motion.— Statement  of  facts  which  appear  to  contradict 
these  conclusions. 

II.  Friction.— The  theory  which  explains  the  excess  of  the  work  of  the  motive  forces  over 
the  useful  work,  by  the  work  of  friction,  is  untenable.— Friction  generates  heat. 

III.  Radiant  heat.— The  nature  of  heat.— Heat  is  living  force. 

lA'.  The  heat  developed  by  friction  is  the  equivalent  of  the  excess  of  the  work  of  the  motive 
forces  over  the  useful  work.— Experiments  of  Joule.— First  definition  of  the  idea  :  Mechanical 
equivalent  of  heat. 

V.  Upon  the  steam-engine.— The  work  of  the  molecular  forces  in  this  engine  is  zero.— Origin 
of  its  motive  power:  Disappearance  of  a  qnantitj'  of  heat  equivalent  to  the  work  performed. — 
Experiments  of  Hirii.— New  determination  of  the  mechanical  equivalent  of  heat. 

YI.  General  proof  and  statement  of  the  fundamental  principle  of  the  equivalence  between 
heat  and  mechanical  work,  or  living  force.— 'I'his  principle  renders  a  complete  revision  of  science 
necessary. — Character  and  scope  of  this  revision. 

VII.  Investigation  of  the  action  of  hear  upon  bodies, — Inner  work,  outer  work. — New  theory 
of  latent  heat.— It  is  an  error  to  compare  the  latent  heat  wiili  the  outer  work,  or  witli  an  incom- 
plete expression  of  the  outer  work.— In  the  present  state  of  science  the  inner  work  eludes 
determination.— Methods  which  enable  us  to  avoid  this  difficulty  and  to  establish  equations 
between  the  mechanical  and  thermal  properties  of  bodies. 

VIII.  Special  investigation  of  gases.— Pacts  which  seem  to  prove  that  the  efEect  of  mo- 
lecular attraction  in  such  bodies  is  imperceptilile.— Conscciuences  :  1.  New  theory  of  the  con- 
stitution of  gases.  2.  In  change  of  volume  there  is  no  inner  work.— Experimental  confirmation 
of  these  consequences  by  Joule.- -Notice  of  the  contradiction  which  appears  to  exist  between 
the  experiments  of  Joule  and  known  properties  of  gases.— Various  forms  of  these  experiments. 
—Deduction  of  the  formula  which  expresses  the  mechanical  equivalent  of  heat  in  terms  of  the 
two  specific  heats,  the  coeftlcient  of  expansion  and  the  volume  of  the  unit  of  weight. 

IX.  Limitation  of  perfect  gases. — In  those  gases  which  do  not  follow  Marriotte's  law,  the 
inner  work  is  perceptible  although  very  small.— Experiments  of  Joule  and  W.  Thompson.— 
Deductions  from  these  experiments. 


LECTUEE  I. 


I. 

Theemodynamics,  or  the  Mechanical  Theory  of  Heat,  is  tlaat  science 
wliicli  treats  of  the  mechanical  effects  of  heat,  and.  of  those  mechanical  pro- 
cesses by  which  heat  is  generated.  This  science  is  yet  in  its  infancy.  It  is 
not  more  than  40  years  ago  *  that  Sadi  Carnot  pointed  out  its  first  problems, 
and  scarcely  30  years  have  elajDsed  *  since  Julius  Eobert  INIayer  indicated  the 
methods  by  which  their  solution  might  be  attempted. 

Nevertheless,  this  science  has  already  reached  a  great  development,  and 
has  attained  points  of  contact  with  almost  all  the  other  sciences.  We  shall 
endeavor,  in  what  follows,  to  obtain  a  comprehensive  view  of  this  rapid  ]3rogress. 

The  new  science  rests  ujDon  a  few  fundamental  principles  of  mechanics,  and 
to  these  let  us  first,  for  a  moment,  direct  our  attention. 

The  law,  according  to  which  the  velocity  of  any  material  point  acted  upon 
by  a  constant  force  changes,  is  well  known  (it  is  «=  |/3  gh).  We  also  know 
that  the  square  of  the  velocity  attained  in. any  given  time  is  equal  to  twice  the 
product  of  the  moving  force  and  the  distance  passed  over,  divided  by  the  mass 

of  the  moving  point  (or  t-'-  =  ""        | . 

The  velocity  increases  or  is  accelerated  when  the  moving  force  acts  in  the 
direction  of  the  original  velocity  of  the  point,  and  it  is  retarded  when  the  mov- 
ing force  acts  in  the  opj)osite  direction. 

The  product  of  the  force  into  the  distance  passed  over,  we  call  the 
"mechanical  effect"  or  "  wokk"  of  the  force.  We  call  this  work  posi- 
tive or  negative,  according  as  the  force  causes  motion  or  opposes  motion  of  the 
point,  i.e.,  according  as  it  acts  in  the  direction  of  the  initial  velocity  or  the 
reverse. 

We  call  the  half  product  of  the  mass  and  square  of  the  velocity  (A  Mv"^)  the 
"VIS  vrvA"  or  "living  foece,"  and  by  the  aid  of  these  two  definitions  we 
may  express  the  foregoing  principle  in  the  following  manner  : 

When  a  iocly    moves  witli  imiformly  accelerated  or  retarded  motion,  the 

*  The  two  introductory  lectures  whicli  follow  were  delivered  by  Prof.  Verdet  before  the 
Chemical  Society  of  Paris,  in  the  year  1862.  The  dates  above  should  therefore  now  be  58  and  48  re- 
speclively.  As  a  popular  and  yet  scientific  exposition  of  the  subject,  these  lectures  are  still  unri- 
valed, and  to  the  beginner  who  desires  to  get  clear  general  ideas  of  the  scope  and  spirit  of  the 
science  they  will,  it  is  thought,  prove  both  interesting  and  valuable,  and  render  the  proper  com- 
prehension of  the  technical  discussion  which  follows  much  easier. 

3 


4  LECTURE  I. 

"  iDorJc "  daring  any  given  tims,  performed  upon  the  body  or  performed  hy  the 
'body,  is  equal  to  the  change  in  the  "  living  force  " — [i.e.,  Work  =  ^  M{  Fo^  —  «i  ^)]. 

This  principle,  whiclx  follows  directly  from  the  above  definitions  and  prin- 
ciples, enables  us  to  measure  forces  by  the  resulting  velocities,  and  may  be 
easily  generalized. 

Thus,  by  the  aid  of  the  Calculus,  we  may  remove  the  limitation  as  to  con- 
stant force,  which  we  have  introduced  for  the  sake  of  clearness.  Then  the 
limitation  as  to  direction  may  be  removed  by  finding  the  component  of  the  in- 
clined force  in  the  direction  of  the  motion,  and  taking  its  work.  Finally,  we 
may  consider  any  system  whatever  of  forces  and  bodies,  and  show  that  in  all 
cases  the  sum  of  the  works  performed  in  ccny  given  time  is  equal  to  the  change  in 
the  sum  of  the  limng  forces  in  that  time.  This  is  the  principle  known  as  the 
equality  of  work  and  living  force,  upon  which  rests  the  entire  theory  of 
machines. 

Mechanical  effect,  or  work,  is  expressed  numerically  by  means  of  a  conven- 
tional unit.  Thus  that  amount  of  work  may  be  taken  as  unity  which  is  per- 
formed in  li  f ting  one  unit  of  weight,  as  one  pound  or  one  kilogram,  against  the 
force  of  gravity,  through  the  vertical  distance  of  one  unit  of  length,  as  one 
foot  or  one  meter.  Work  is  thus  measured  in  "  foot-pounds  "  or  "  meter-kilo- 
grams." If,  thus,  we  say  that  the  work  of  any  system  is  positive  and  equal  to 
100  ft.  lbs. ,  we  mean  that  by  means  of  this  system  we  can  perform  the  same  work 
as  would  be  performed  by  a  weight  of  100  lbs.  descending  through  a  height  of 
one  foot,  or,  regarding  the  force  of  gravity  as  constant,  by  the  descent  of  one 
pound  through  100  feet.  In  like  manner,  100  meter-kilograms  signifies  the  work 
performed  by  the  descent  of  one  kilogram  through  the  distance  of  100  meters. 

Inversely,  a  negative  work  of  100  ft.  lbs.  denotes  an  expenditure  of  work 
by  the  system  equal  to  that  expended  in  raising  a  weight  of  one  pound  through 
100  feet,  or  100  lbs.  through  one  f odt,  the  final  velocity  being  zero. 

It  is  not  our  purpose  to  indicate  here  how  the  entire  theory  of  machines 
follows  from  this  equation  of  work  ;  but  it  is  necessary  to  direct  attention  to 
two  general  conditions  which  the  motion  of  any  machine  must  satisfy,  and 
which  are  expressed  in  this  equation. 

First,  in  every  machine  whose  motion  has  become  constant,  or,  in  general, 
in  any  system  whose  velocity  is  independent  of  the  time,  the  sum  of  the  living 
forces  is  constant,  and  hence  in  any  period  which  we  consider,  the  sum  of  the 
works  zero.  In  other  words,  the  work  of  the  moving  forces  is  constantly  equal 
to  the  work  of  the  resistances,  and  has  a  contrary  sign.  If  the  velocities  indeed 
are  not  constant,  but  periodic  in  their  variation,  as,  for  example,  is  the  case  in  a 
machine  with  a  reciprocating  motion,  then,  although  equality  no  longer  exists 
between  the  work  of  the  moving  forces  and  of  the  resistances  for  any  arbitrary 
interval  of  time,  still  it  does  exist  for  the  duration  of  a  full  period,  or  for  any 
entire  number  of  such  periods. 

If,  further,  the  forces  which  act  upon  a  system  have  at  one  time  an  effect 
opposed  to  the  action  of  the  individual  points  of  this  system  upon  each  other,  if 
therefore  they  act  in  the  directions  of  the  lines  joining  these  points,  and  are 
dependent  only  upon  the  distances  apart  of  the  points,  and  if  at  another  time 
they  proceed  from  a  center,  and  are  subject  to  the  same  conditions  ;  then,  the 
sum  of  the  living  forces  is  the  same  both  at  the  beginning  and  end  of  a  time 
such  that  the  bodies  of  the  system  return  to  their  first  positions.     The  sum  of 


INTBODVGTION.  5 

the  works  of  the  forces  during  this  time  is  zero.     These  conditions  are  satisfied 
by  every  case  which  occurs  in  nature. 

This  law,  which  rests  upon  the  surest  ideas  which  we  have  concerning-  the 
operation  of  the  forces  of  nature,  is  nothing  more  than  the  principle  of  the 

IMPOSSIBILITY   OF  PERPETUAL   MOTION. 

According  to  this  it  is  impossible  by  any  combination  of  natural  forces  to 
make  a  machine  whose  parts  being  once  set  in  motion  and  then  left  to  the 
operation  of  gravity,  or  other  similar  forces,  and  their  own  mutnal  action,  shall 
later  return  to  their  original  positions  with  greater  velocities  than  they  at  first 


A  perpetual  motor  means,  then,  a  machine  which,  being  put  in  motion  and 
left  to  itself  will,  in  a  certain  time,  regain  its  original  velocity,  and  at  the  same 
time  impart  to  some  body,  originally  at  rest,  a  certain  velocity.  It  is  clear  that 
both  cases  of  impossibility  are  identical.* 

It  does  not  appear  at  sight  easy,  proceeding  from  these  principles,  to  make 
any  new  discoveries.  The  theory  of  simple  machines  is  firmly  founded,  and 
all  analyses  of  deceptive  discoveries  of  a  perpetual  motion  are  to-day  completely 
devoid  of  interest.  Nevertheless,  it  is  from  a  new  application  of  these  appar- 
ently thoroughly  explored  principles  that  the  entire  mechanical  heat  theory  has 
arisen. 

It  will,  for  our  purposes,  be  sufficient  to  give  heed  to  these  two  rules  : 

First,  always  to  recognize,  together  v/ith  the  outward  and  visible  motions  of 
any  machine,  those  less  perceptible  interior  motions  of  the  atoms  of  bodies 
"which  escape  observation  by  our  senses. 

Second,  whenever,  following  customary  theories,  we  meet  with  a  force 
whose  mode  of  action  does  not  agree  with  the  general  laws  of  action  of  natural 
forces,  we  must  regard  this  force  as  a  mathematical  fiction,  and  seek  to  estab- 
lish its  true  nature. 

Without  these  two  maxims  every  theory  of  machines  must  lead  astray  ; 
every  machine  in  motion  must  appear  as  a  direct  contradiction  of  the  law  of 
equality  of  the  work  of  the  motive  power  and  the  work  of  the  resistance,  or  as 
a  solution  of  the  problem  of  perpetual  motion.  The  only  means  of  avoiding 
such  contradiction  would  be  to  propound  views  as  to  the  nature  and  mode  of 
action  of  heat,  whose  scope  would  exceed  that  of  the  simple  circle  of  pheno- 
mena which  first  suggested  them. 

II. 

Next,  we  assert  that  in  no  machine  which  has  attained  a  state  of  uniform 
motion  can  the  work  of  the  resistances  be  equal  to  the  work  of  the  moving 
forces.  Although  this  assertion  appears,  in  view  of  the  above,  paradoxical,  yet 
it  simply  expresses  what  at  bottom  can  be  found  in  any  text-book  upon  mechan- 
ics. It  is  nothing  more  than  the  true  interpretation  of  the  preponderance  of 
the  work  of  the  moving  forces  compared  with  that  which  we  call  the  "  useful 
work." 

Let  us  consider  an  hydraulic  machine  which  is  designed  to  raise  water,  i.e., 
to  produce  an  effect  similar  to  that  which  it  receives.     This  will  simplify  the 

*  See  Note  1,  at  the  end  of  these  lectures. 


6  LECTURE  1. 

comparison  of  the  two  kinds  of  work.  In  a  machine  of  this  kind  a  certain 
quantity  of  water  enters  in  a  given  time,  falls  through  a  certain  distance  per- 
forming work,  and,  if  the  machine  is  perfect,  departs  with  the  velocity  which 
it  had  ])efore  falling.  The  product  of  the  weight  of  water  and  the  height  of 
fall  is  evidently  the  work  of  tho  moving  force.  In  the  same  time,  the  machine 
takes  a  certain  quantity  of  water  from  some  reservoir,  it  may  be  from  the  very 
stream  which  furnishes  the  motive  water  itself,  and  raises  it  up  to  another  and 
higher  reservoir.  This  negative  work,  against  gravity,  is  the  product  of  the 
weight  of  water  raised,  into  the  difference  of  level  between  the  two  reservoirs, 
and  is  what  we  call  the  "  useful  icorJc.'" 

Now,  every  one  knows  that  this  useful  work  is  only  a  fraction  of  the  work 
of  the  moving  force.  This  fact  is  ordinarily  accounted  for  by  the  considera- 
tion of  what  we  call  the  "  passive  resistance,"  that  is,  by  the  assumption  of 
forces  which  oppose  the  motion  of  the  machine,  and  thus  perform  a  negative 
work  exactly  equal  to  the  excess  of  the  work  of  the  moving  force  above  the 
useful  work.     Let  us  tee  what  value  this  explanation  has. 

There  is  one  part  of  the  passive  resistance  of  somewhat  indefinite  amount. 
To  this  belong  every  contribution  of  velocity  to  surrounding  bodies,  either  to 
the  air  or  to  the  foundations,  which  theory,  of  course,  assumes  as  firm.  All 
these  constitute  a  useless  development  of  living  force  at  the  expense  of,  and 
equivalent  to,  a  certain  fraction  of  the  work  of  the  moving  force.  In  by  far 
the  greater  number  of  cases,  however,  these  constitute  the  least  portion  of  the 
work  of  the  passive  resistances.  Much  the  greatest  portion  must  nearly  always 
be  attributed  to  the  action  of  a  certain  special  force  which  bears  the  name  of 
"  friction,"  and  to  this  force  we  now  wish  to  call  special  attention. 

What,  then,  is  friction  1  It  is  purely  a  resisting  force,  incapable  of  causing 
motion  in  the  machine,  or  of  increasing  its  velocity.  It  is  a  force  which  always 
tends,  when  two  surfaces  in  contact  move  with  different  velocities,  to  diminish 
the  velocity  of  the  fastest. 

It  is  not  an  elementary  work,  but  the  result  of  actions  which  occur  between 
the  molecules  of  the  rubbing  surfaces.  We  know  little  or  nothing  of  these 
actions,  more  than  that  they  must  obey  the  general  laws  which  we  have  laid 
down  just  now,  while  speaking  of  perpetual  motion.  We  do  not  need,  how- 
ever, to  know  anything  whatever  about  them,  in  order  to  demonstrate  that 
they  can  furnish  no  work,  and  hence  can  give  no  information  as  to  the  facts  to 
be  accounted  for.  In  machines,  ordinarily,  rubbing  surfaces  are  ground  down, 
also  the  lubricating  materials  undergo  a  change.  We  might  suppose  that  the 
work  corresponding  to  such  molecular  changes  was  the  exact  equivalent  of  that 
portion  of  the  excess  of  the  work  received  over  that  performed,  which  we 
ascribe  to  friction.  But  it  is  easy  to  conceive  of  a  machine  whose  rubbing  sur- 
faces are  so  smooth  and  of  such  hard  material  as  not  to  rub  down  perceptibly 
in  a  long  time  ;  it  would  not  indeed  be  difficult  to  practically  realize  such  a 
machine.  If  we  consider,  in  such  a  case,  the  work  of  molecular  forces,  which 
is  the  cause  of  the  friction,  during  the  period  between  two  precisely  identical 
positions  of  the  machine,  it  is  at  once  evident  that  this  work  must  be  zero,  be- 
cause at  the  beginning  and  end  of  the  period  the  relative  positions  of  the  mole- 
cules is  the  same. 

Where,  then,  does  the  ordinary  explanation  of  the  excess  of  the  work  received 
over  the  useful  work  lead  us  ?    Can  we  recognize  in  it  anything  else  than  a 


INTBOBTJGTION.  7 

pure  mathematical  fiction,  which  may  perhaps  be  useful  as  a  temporary  pres- 
entation of  an  unltnown  process,  but  which  must  be  rejected  by  every  one  not 
jjrepared  to  deny  the  most  certain  conclusions  of  science  ?  Must  we  not  con- 
clude, that  in  every  case  where  we  have  friction  without  change  of  surface, 
there  must  be  some  unobserved  change,  which  is,  in  fact,  equivalent  to  work, 
and  which  seems  to  absorb  the  friction  ? 

To  the  eye  of  the  pure  mechanic  no  such  change  may  be  apparent,  the 
physicist,  however,  without  doubt,  will  recall  a  well-known  phenomenon, 
familiar  even  to  ordinary  experience,  and  which  has  already  more  than  once 
been  the  subject  of  scientific  investigation.  I  speak  of  the  increase  of  tempera- 
ture which  always  takes  place  when  surfaces  are  rubbed,  and  which  is  more 
considerable  the  greater  the  friction  ;  or,  what  amounts  to  the  same  thing,  the 
greater  the  unexplained  loss  cf  work. 

Without  pausing  to  recapitulate  the  laws  of  this  phenomenon,  let  us  direct 
attention  to  its  essential  character.  It  is-  a  heating  which  corresponds  to  no 
cooling  of  any  other  part  of  the  machine.  It  is  not  another  distribution  of  heat 
which  already  existed,  but  it  is  a  generation,  or,  still  better,  an  actual  creation 
of  heat.  What  is  more  natural  than  to  recognize  in  this  the  equivalent  of  the 
excess  of  the  work  received  over  the  useful  work,  which  we  are  otherwise  at  a 
loss  how  to  account  for  ? 

III. 

In  order  to  estimate  the  value  of  this  supposition,  let  us  consider  a  kind  of 
action  entirely  different  from  that  which  takes  place  in  machines — viz.,  the 
phenomena  of  radiant  heat.  Let  us  recall  the  experiments  of  Delaroche,  Be- 
rard,  Melloni,  Knoblauch,  Tyndall,  and  other  physicists,  upon  what  are  called, 
both  in  popular  and  in  scientific  parlance,  "  heat  rays." 

These  experiments  are  in  complete  accord  with  those  by  which  the  true 
nature  of  light  is  revealed,  and  as  the  view  held  to-day  as  to  the  nature  of 
light  is  held  by  all,  we  are  in  like  manner  forced  to  conclude  that  heat  rays  are 
nothing  else  than  a  certain  vibratory  motion  of  the  ether  of  space.  Thus,  in 
accordance  with  experiment,  we  say,  that  if  a  body  is  brought  near  to  another 
of  lower  temperature,  certain  vibrations  are  generated,  which  follow  certain 
laws.  Upon  these  vibrations  depend  the  phenomena  of  the  imparting  of  heat, 
and  under  certain  circumstances  they  are  capable  of  acting  upon  our  eyes  so  as 
to  give  rise  to  the  phenomena  of  light  also.  We  have  no  reason  to  suppose 
that  the  two  kinds  of  phenomena  have  different  causes. 

This  fundamental  identity  of  radiant  heat  and  light  was  demonstrated  twenty 
years  ago  *  by  Melloni  in  his  paper  "  Upon  the  Identity  of  Rays  of  all  Kinds," 
read  before  the  Academy  at  Naples,  February  2,  1842.  Still,  Melloni  recog- 
nized that  an  important  step  remained  to  be  made  to  complete  the  demonstra- 
tion. The  interference  of  heat  rays  had  not  yet  been  experimentally  shown  ; 
no  one  had  yet  succeeded  by  two  rays  of  heat  in  producing  cold,  as  by  two 
rays  of  light,  under  proper  conditions,  darkness  had  been  caused.  Five  years 
later,  Fizeau  and  Foucault  detailed,  in  a  paper  before  the  Academy,  experi- 
ments by  which  the  interference  of  heat  was  made  as  evident  as  that  of  light. 
[Comptes  rendus,  Vol.  XXV.,  and  Poggend.  Annalen,  Bd.  73.) 

*  These  lectures  were  delivered  by  Verdet  in  1862. 


8  LECTURE  I. 

After  this,  not  a  single  doubt  remained  to  oppose  to  a  theory  which  recog- 
nized in  the  heat  rays  a  system  of  vibrations.  We  shall  consider  it  as  an 
undoubted  fact,  that  in  a  body  which  is  brought  to  a  higher  temperature, 
vibrations  of  its  molecules  are  caused  ;  in  other  words,  the  body  contains  a 
certain  amount  of  living  force.  While  anuther  body  of  higher  temperature, 
which  serves  as  a  source  of  heat,  causes  this  development  of  living  force,  it 
cools  gradually  itself. 

Inversely,  when  the  vibrations  which  constitute  a  system  of  heat  rays  meet 
a  body  which  possesses  the  property,  as  we  say,  of  "absorbing  heat,"  and 
become  diminished  or  disappear,  the  body  becomes  heated. 

The  cooling  of  a  body  by  radiation,  therefore,  corresponds  to  the  generation 
in  other  bodies  of  a  certain  amount  of  living  force  ;  the  heating  of  a  cold  body 
by  the  absorption  of  radiant  heat,  on  the  other  hand,  corresponds  to  a  diminu- 
tion of  living  force  in  others.  Heating  and  cooling,  therefore,  are  phenomena 
of  the  same  kind,  whatever  may  be  their  cause.  They  must,  in  all  cases,  be 
considered  as  pure  mechanical  operations.  Heating  can  only  be  the  sum  total 
of  those  changes  which  take  place  during  the  disappearance  of  a  certain  amount 
of  living  force  ;  i.e.,  either  a  performance  of  work  or  a  development  of  living 
force,  or  a  combination  of  both.  It  is  evident  that  heating  corresponds  to 
mechanical  work. 

Heat  acts  to  change  the  volume  of  bodies,  the  molecules  are  forced  farther 
apart  against  their  forces  of  attraction,  and  thus  a  negative  work  is  perfo2-med. 
At  the  same  time  occurs  that  change  of  the  properties  of  the  body  which  we 
call  rise  of  temperature,  and  it  is  easy  to  see  in  it  the  effect  of  the  change  in 
the  sum  of  the  living  forces  of  the  molecules. 

It  makes  very  little  difference  whether  we  accept  or  reject  these  last  conclu- 
sions ;  it  is  none  the  less  certain  that  the  heating  of  a  body  represents  a  certain 
performance  of  work  and  the  development  of  a  certain  amount  of  living  force, 
or  still  better,  is  such  force.  The  work  in  question  consists  of  molecular  dis- 
turbances Avhich,  indeed,  escape  observation,  and  are  only  visible,  finally,  in 
the  change  of  form  and  dimensions  of  the  body  ;  the  living  force  is  also  as 
difficult  of  direct  observation,  and  consists  neither  of  the  motion  of  the  body  as 
a  whole,  nor  of  directly  visible  motions  of  its  parts,  such,  for  example,  as  con- 
stitute sound  phenomena.  It  consists,  in  all  i^robability,  in  vibrations  of  the 
smallest  particles  of  matter,  and  eludes  our  senses.  Considered  from  a  me- 
chanical standpoint,  these  speculations  have  no  importance,  and  cannot  prevent 
our  recognizing  in  the  heating  of  a  body,  mechanical  work,  just  as  plainly  and 
as  certainly  as  in  the  raising  of  a  weight  or  the  motion  of  a  projectile. 


IV. 

Let  us  return  now,  in  the  light  of  these  new  principles,  to  the  consideration 
of  the  machine  which  we  have  instanced,  and  those  questions  which  then  arose 
will  now  find  immediate  solution.  The  heat  which  is  developed  at  those  points 
where  friction  occurs  is  a  mechanical  phenomenon,  a  combination  of  mechanical 
work  and  living  force  in  a  relation  which  we  shall  determine  more  pi-ecisely 
hereafter.  It  is  evident  that  this  heat  may  be  equivalent  to  the  difference 
which  exists  between  the  work  of  the  motive  forces  and  the  useful  work,  the 
explanation  of  which  we  have  been  seeking.     I  say  may  he,  and  you  will,  per- 


INTRODUCTION.  9 

haps,  be  inclined  to  add,  must  he.  The  equation  of  work  must  necessarily  be 
satisfied  at  every  instant,  only  we  must  not  limit  it  to  those  living  forces  or 
visible  effects  which  are  usually  alone  considered,  but  we  must  also  include 
those  living  forces,  or  that  work,  which  we  know  in  the  shape  of  heat.  If  we 
neglect  this  term  of  the  equation  of  works,  the  fundamental  theorem  of  applied 
mechanics  may  indeed  appear  incorrect,  but  by  its  introduction  all  difficulties 
disappear. 

Having  now  arrived  at  this  point,  we  can  submit  the  correctness  of  our 
conclusions  to  the  test  of  experiment.  We  may  seek,  for  instance,  whether  it 
is  really  true  that  the  heat  generated  by  friction  in  a  machine  is  exactly 
equivalent  to  the  unexplained  difference  between  the  useful  work  and  that 
received.  Although,  indeed,  it  may  be  impossible  to  measure  this  quantity  of 
heat  in  the  condition  of  living  force  or  work,  as  we  measure,  for  instance,  the 
work  of  gravity  upon  a  body  of  one  pound  weight  which  falls  through  one 
foot,  still  we  may  measure  it  relatively  by  comparison  with  another  quantity  of 
heat,  which  may  be  sharply  defined  and  taken  as  unity. 

The  result  of  such  a  procedure  will  give  us  this  quantity  of  heat  expressed 
in  terms  of  these  units,  and  then  if  we  know  the  ratio  of  this  unit  of  heat  to  the 
unit  of  work,  that  is,  the  number  of  foot-pounds  or  meter-kilograms  corre- 
sponding to  each  heat  unit,  we  can  easily  find  the  work  equivalent  to  the  heat, 
which  will  be  the  difference  between  the  useful  work  and  the  work  originally 
imparted.  This  constant  ratio  will  therefore  determine  the  mechanical  value 
of  those  heat  effects  which  we  assume  as  constituting  one  unit  of  heat. 

This  has  been  established  by  experiment.  The  physicist  Joule,  who  has 
perhaps  contributed  more  than  any  one  else  to  the  science  of  Thermodynamics, 
has  investigated  friction  of  various  kinds  in  such  a  way  as  to  determine  the 
amount  of  heat  developed  in  comparison  to  the  work  expended.  He  used  a 
very  simple  mechanism,  which,  by  means  of  a  falling  weight,  set  in  motion  a 
small  paddle-wheel,  which  turned  while  immersed  in  water  or  mercury,  the 
motion  of  the  liquid  being  prevented  by  partitions.  The  friction  of  the  liquid 
particles  upon  each  other,  upon  the  partitions  and  upon  the  paddles,  generated 
a  certain  amount  of  heat,  which  could  be  estimated  from  the  rise  of  tempera- 
ture of  the  various  parts  of  the  apparatus.  The  work  corresponding  to  this 
heat  was  given  by  the  fall  of  the  weight  used,  due  regard  being  had  for  the 
friction  of  those  parts  of  the  machine  out  of  the  liquid.  Thus  was  determined 
the  ratio  of  the  mechanical  work  expended  to  the  heat  produced.  Experiments 
with  water  gave  for  every  unit  of  heat,  that  is,  for  every  kilogram  of  water 
raised  one  degree  Centigrade,  the  equivalent  work  of  424  meter-kilograms. 
(If  we  take  as  the  unit  of  heat,  one  pound  of  water  raised  one  degree  Centi- 
grade, we  have  about  1,390  foot  lbs.  If  we  take  one  pound  of  water  raised  one 
degree  Fahrenheit,  we  have  about  773  foot  lbs.  All  three  equivalents  are  in 
use.) 

Experiments  with  mercury  gave  425  meter-kilograms,  or  very  precisely  the 
same  as  water.  Joule  made  still  a  third  determination,  witb  an  iron  ring  in- 
stead of  a  paddle-wheel,  which  ring  he  caused  to  rub  upon  an  iron  plate  im- 
mersed in  water,  and  found  in  this  case  426  meter-kilograms. 

You  will  doubtless  be  surprised  at  the  close  correspondence  of  these  three 
numbers.  When  I  add  that  each  is  the  mean  of  a  large  number  of  determina- 
tions, it  will  be  readily  confessed,  that  in  this  work  of  Joule,  classic  even  to- 


10  LECTURE  I. 

day,  is  found  the  experimental  verification  of  our  new  principle.  You  will 
admit  tliat  the  mean  of  the  determinations  for  water,  which  are  regarded  as 
the  most  reliable,  or  424,  represents  with  tolerable  exactness  the  quantity  of 
work  which  is  the  equivalent  of  that  living  force  among  the  particles  of  a  body, 
to  which  we  gipe  the  name  of  "  one  heat  unit."  Let  us  pause  a  moment  to 
consider  the  significance  of  this  number.  It  expresses,  that  from  a  mechanical 
standpoint,  we  produce  two  equivalent  effects,  whether  we  generate  one  unit 
of  heat,  or  whether  we  raise  424  kilograms  through  one  meter.  In  other 
words,  in  every  application  of  the  equation  of  works,  in  which  we  take  account 
of  both  the  living  force  of  heat,  and,  at  the  same  time,  of  the  work  of  the  visi- 
ble forces,  we  must,  for  every  unit  of  heat,  add  424  units  to  the  negative  work 
or  to  the  living  forces.  This  relation  is  independent  of  the  special  method  of  the 
production  of  heat  by  friction.  It  follows  from  principles,  whose  generality 
has  already  been  proved,  that  424  can  be  in  every  case  regarded  as  the  mechan- 
ical EQUIVALENT  OF  HEAT. 

If  it  should  possibly  appear  rather  premature  to  consider  this  definite 
numerical  value  as  absolutely  correct,  still  no  objections  can  be  urged,  and  no 
doubt  remain  as  to  the  entire  correctness  of  the  principle  of  the  equivalence  of 
work  and  heat ;  for,  making  allowance  for  errors  unavoidable  in  all  experi- 
ments, the  most  diverse  determinations  all  agree  in  giving  us  the  same  value. 


We  find  the  first  confirmation  of  Joule's  experiments  in  the  researches  of 
Favre  upon  the  friction  of  steel  on  steel  :  but  we  shall,  for  a  moment,  pass 
over  such  confirmations,  in  order  to  direct  attention  to  still  another  contradic- 
tion which  seems  to  exist  between  the  usual  theory  of  machines  and  general 
mechanical  laws,  which  is,  in  a  certain  sense,  the  opposite  of  the  preceding, 
and  which  only  disappears  when  we  apply  those  principles  which  we  have 
already  deduced.  It  can  be  easily  shown,  that  if  we  depart  from  these  princi- 
ples, every  machine  which  is  moved  by  heat  can  be  regarded  as  a  perpetual 
motion,  which  continually  generates  living  force  in  surrounding  bodies,  with- 
out any  change  in  its  own,  without.  In  fact,  a  positive  work  of  the  motive  forces 
equivalent  to  the  living  forces  generated. 

Let  us  take  as  an  example  that  most  important  and  well-known  machine  of 
our  civilization — the  steam-engine.  Consider,  then,  with  me  a  machine  which 
has  arrived  at  the  condition  of  its  normal  activity,  and  in  order  to  fix  our  ideas, 
let  us  take  a  condensing  engine.  What  takes  place  during  one  revolution  or 
one  double  stroke  of  the  piston?  A  certain  quantity  of  water  of  low  tempera- 
ture is  forced  by  the  feed-pump  into  the  boiler,  it  is  there  heated  and  converted 
into  saturated  steam  of  a  temperature  above  100°  C.  The  water  in  this  new 
condition  enters  the  cylinder,  raises  the  piston,  expands  in  volume,  and  finally 
returns  to  the  condenser,  where  it  retakes  its  original  condition,  viz.,  water  of 
low  temperature.  We  have  thus,  at  the  end  of  this  cycle  of  changes,  every- 
thing in  the  same  condition  as  at  first.  Not  only  are  all  parts  of  the  machine 
in  the  same  relative  position,  but  the  moving  agent  also  has  returned  to  pre- 
cisely its  original  condition.  (The  amount  of  water  which  is  injected  into  the 
condenser,  in  order  to  condense  the  steam,  need  cause  no  confusion  ;  this  water 
is  simply  a  cooling  agent,  which  might  be  replaced  by  any  other  without  cliang- 


INTRODUCTION.  11 

ing  tlie  operation  of  tlie  machine.  Tbus,  for  example,  we  may  have  for  the 
condeuser  a  spiral  pipe  immersed  in  cold  water,  and  which,  therefore,  coutains 
only  that  amount  of  water  which  is  used  in  one  stroke  of  the  piston.)  In  such 
a  case,  it  is  at  once  evident  that  at  the  beginning  and  eud  of  each  period  of  the 
machine,  the  conditions  both  of  the  fluid  motor  and  of  the  mechanism  are  pre- 
cisely the  same,  and  we  can  at  once  conclude  that  the  sum  of  the  works  during 
that  period  within  the  machine  must  be  zero.  This  separation  of  the  motive 
and  cooling  fluids  is,  in  fact,  actually  accomplished  in  those  engines  worked  by 
ether  or  chloroform  steam  ;  it  would  be  in  like  manner  allowable,  in  principle 
at  least,  in  ordinary  steam-engines.  The  motive  work  of  the  steam  is  com- 
puted, like  the  work  of  friction,  by  an  empirical  expression  for  an  imperfectly 
known  fact.  In  reality,  the  sttm  of  the  works  of  all  the  elementary  forces,  i.e., 
the  work  of  the  mutual  actions  exerted  by  the  molecules  of  the  liquid,  of  tlie 
steam  and  of  the  parts  of  the  machine,  are  equal  to  zero.  And  yet  the  machine 
ic  continually  imparting  living  force  to  exterior  bodies,  raising  weights,  shaping 
metals,  in  short,  performing  work.  Perpetual  motion  seems  accomplished. 
The  outer  Avork  of  the  machine  does  not  seem  to  correspond  either  to  an  equiv- 
alent work  within  the  machine,  nor  yet  to  a  disappearance  of  living  force. 

Such,  at  least,  is  the  state  of  things  so  long  as  we  recognize  in  the  steam- 
engine  purely  mechanical  processes  only  ;  so  long,  at  least,  as  we  do  not  search 
for  other  living  forces  than  those  possessed  by  the  visible  portions  of  the  ma- 
chine. The  difficulty,  however,  vanishes  as  soon  as  we  take  into  account  the 
living  force  of  the  heat.  During  the  action  of  the  machine,  the  steam  gener- 
ated takes  away  heat  with  it  from  the  boiler,  and  gives  up  heat  in  the  con- 
denser, where  it  becomes  water  again.  If  these  two  quantities  of  heat  were 
equal,  the  difficulty  already  noticed  would  still  hold  in  full  force.  If,  however, 
they  are  not  equal  ;  if  the  quantity  of  heat  received  by  the  condenser,  or  carried 
away  by  the  cooling  water,  is  less  than  that  furnished  by  the  boiler,  then  the 
difficulty  is  solved.  The  disappearance  of  a  certain  quantity  of  heat  during 
the  cycle  of  changes,  corresponds,  in  fact,  according  to  our  new  principle,  to 
the  disappearance  of  a  certain  amount  of  living  force. 

In  the  same  time  in  which  outer  work  is  performed  by  the  machine,  or  liv- 
ing force  is  developed,  an  equivalent  quantity  of  living  force  disappears  within 
the  machine,  and  the  general  laws  of  mechanics  hold  good. 

In  order  to  confirm  this  conclusion  experimental  proof  is  necessary.  We 
must  measure,  on  the  one  hand,  the  work  of  the  machine,  and  on  the  other, 
the  loss  of  heat  within  the  machine,  and  then,  if  our  conclusions  are  correct, 
there  must  be  found  between  the  two  a  certain  constant  relation. 

The  necessity  for  the  existence  of  such  a  constant  relation  will  be  evident, 
without  repeating  the  considerations  which  in  the  case  of  friction  led  us  to  a 
similar  conclusion. 

For  every  unit  of  heat  which  disappears  in  the  machine,  it  must  furnish 
434  units  of  outer  work,  or  it  must  generate  an  equal  quantity  of  living  force. 

The  experiment  is  difficult,  much  more  so  than  the  experiments  of  Joule 
upon  friction  ;  but  it  has  been  performed,  and  without  going  into  details,  I 
.will  endeavor  to  point  out  the  various  operations  which  compose  it.  In  an  en- 
gine whose  motion  has  become  constant,  the  quantity  of  steam  used  for  a  cer- 
tain number  of  strokes  is  measured  ;  the  physical  condition  of  this  steam  as  it 
enters  the  cylinder  from  the  boiler  is  exactly  determined  by  measuring  its  tem- 


12   .  LEGTUBE  I. 

pei-ature  aud  pressure  ;  we  also  so  arrange  that  it  shall  enter  the  cylinder  with- 
out carrying  with  it,  mechanically  suspended,  any  appreciable  amount  of  watery 
particles,  and  without  being  heated  above  its  point  of  saturation.  All  these 
conditions  being  satisfied,  we  have,  in  connection  with  our  knowledge  of  the 
total  heat  required  for  vaporization  (made  known  by  the  experiments  of  Reg- 
nault),  all  the  data  which  we  need  in  order  to  calculate  the  amount  of  heat 
used  in  a  given  time  in  order  to  convert  the  water  in  the  condenser  into  steam. 

On  the  other  hand,  we  can  find,  without  great  difficulty,  the  quantity  of 
heat  which  in  the  same  time  is  given  up  in  the  condenser.  It  is  sufBcient  to 
determine  the  quantity  of  water  used  for  cooling,  which  is  neces^sary  to  pre- 
serve the  temperature  of  the  condenser  constant  in  spite  of  the  continual 
admission  of  steam,  and  also  the  temperature  of  the  condenser  and  that  of 
the  reservoir  from  which  the  cooling  water  is  taken. 

Thus,  in  the  first  case,  if  the  water  to  be  converted  into  steam  enters  the 
boiler  with  the  temperature  of  0",  and  if  T  is  the  temperature  of  the  steam, 
we  have,  according  to  Eegnault's  experiments,  for  the  formation  of  each  unit 
in  weight  of  steam, 

606.5  +  0.305r° 

heat  units.  But  the  water  comes  from  the  condenser  where  the  temperature 
is  f;  the  heat  received,  therefore,  is  less  than  that  which  is  necessary  to  raise 
each  unit  in  weight  of  water  from  0^  to  T'  by  the  amount  which  Is  necessary 
to  raise  it  from  0°  to  t° ,  or  by  t  heat  units.  This  is,  regarding  the  specific 
heat  of  the  water  as  constant,  tolerably  exact  for  those  limits  of  temperature 
which  the  condenser  never  exceeds. 

In  the  second  case,  let  t°  be  the  temperature  of  the  condenser,  and  0  the 
temperature  and  p  the  weight  of  the  injection  water  in  a  given  time.  The 
heat  given  out  by  the  concentration  of  the  steam  must  be  equal  to  that 
absorbed  by  the  injection  water.  This  heat  is,  then,  that  which  is  necessary 
to  raise  p  units  weight  of  water  from  G°  to  f,  or  it  is  equal  to 

p{r -()'). 

The  calorimetric  part  of  the  experiment  is  completed  by  determining  and 
adding  the  losses  of  heat  due  to  conduction,  radiation,  and  disturbances  of  the 
air.  The  most  difficult  part  of  the  experiment  is  the  mechanical.  In  deter- 
mining the  total  work  of  the  machine  we  cannot  use  the  friction  brake.  If  we 
do,  we  shall  only  determine  the  useful  work,  to  which  we  should  then  have  to 
add  the  work  absorbed  by  the  passive  resistances,  which  latter  are  almost 
impossible  of  exact  determination.  We  must  therefore  adopt  an  entirely 
different  method.  Thus,  by  means  of  the  steam  indicator,  we  can  find  the 
mean  pressure  of  the  steam  upon  the  piston,  and  then,  knowing  the  length  of 
stroke,  can  determine  accurately  the  total  work  performed.* 

*  Thus  we  may  conceive,  instead  of  the  actual  pressure  of  the  steam  upon  the  piston,  a 
sinking  weight  and  pulley  so  arranged  that  the  motion  of  the  machine  is  unchanged.  The 
work  of  the  falling  weight  is  that  which  we  call  the  motive  work  of  the  machine,  and  is  exactly 
that  which  we  obtain  by  the  measurement  in  the  text.  In  reality  there  is  a  complete  compen- 
sation between  the  positive  and  negative  work  of  the  forces  in  the  machine.  The  rising  of  the 
piston  is  a  constant  action,  and  the  mechanism  requires  for  this  action  a  force  of  definite  inten- 
sity. We  can,  therefore,  without  contradiction,  continue  to  speak  of  the  work  of  the  steam  in 
the  machine.  We  shall  further  see  that,  very  probably,  the  steam  raises  the  piston  by  imparting 
a  portion  of  the  living  force  of  its  molecules.    See  Note  11,  at  the  end  of  these  lectures. 


INTRODUCTION. 


13 


The  necessity  of  determining  successive  values  of  a  pressure  whicli  varier 
rapidly,  prevents  the  use  of  the  ordinary  manometric  apparatus,, by  which  iho 
elastic  force  of  steam  may  be  measured  with  almost  absolute  exactness.  We 
have  to  make  use  of  the  steam  indicator.  In  spite  of  the  inaccuracies  of 
this  instrument,  which  was  constructed  to  meet  practical  needs  rather  than 
those  of  scientific  investigation,  the  results  obtained  answer  unmistakably 
the  question  at  issue.* 

This  long  and  laborious  work,  the  chief  steps  of  which  we  have  thus 
detailed,  has  been  successfully  performed  by  Hirn,  who  knew  how  to  make 
use  of  a  large  factory  for  the  solution  of  an  abstract  question  of  science. 
His  measurements  were  made,  not  uijon  a  miniature  model  in  a  scientific 
collection,  nor  in  the  laboratory  ;  but  upon  engines  of  100  and  200  horse- 
power, and  in  the  very  halls  of  industry.  These  circumstances  had  two 
especial  advantages.  On  the  one  hand  they  met  the  objections  of  practical 
men,  who  are  disposed  to  regard  with  distrust  what  they  call  "  cabinet  experi- 
ments ;  "  on  the  other  hand,  and  this  is  still  more  important,  by  reason  of 
the  large  dimensions  of  apparatus  and  the  long  duration  of  the  experiments, 
those  thousand  accidental  disturbances  which  ever  attend  new  discoveries 
could  be  eliminated. 


*  The  indicator  consists  ot  a  small  cj'linder  C,  Fig.  1,  within  which  moves  the  piston  K, 
attached  to  the  spring/.  The  space  below  the  piston  communicates  with  the  cylinder  of  the 
engine.  As  the  pre^'^ure  increases  or  diminishes,  the  piston  vi«es  or  falls  \  pencil  G  which 
partakes  of  the  motion  of  the  piston,  describes  \  line  npon  a  strip  of  paper  ^\ound  aiound 
the  drnm  G.  Thib 
drum  is  made  to  re 
volve  about  its  ver 
tical  axis  by  means 
of  acordandpnllej 
The  cord  leads  to 
the  cross-head  or 
some  reciprocating 
part  connected  with 
the  piston.  By 
means  of  a  spring 
within  the  drum,  it 
is  made  to  revohe 
back  during  the  re 
turn  stroke.  The 
pencil  f?  thus  traces 
a  closed  curve,  the 
length  of  which  rep 
resents  the  stroke, 
and  the  varying 
height  of  which  rep 
resents  thepressuie 
at  corresponding 
positions  of  the  pis- 
ton. If  the  piston 
were  without  fric- 
tion, the  area  of  this 
curve,  measured 
from  the  straight  line  described  when  the 


Fig.  1. 


ressure  on  both  sides  of  the  piston  is  constant  and 
equal  to  the  atmospheric  pressure,  will  be  proportional  to  the  total  disposable  work.  We  see 
that  it  is  impossible  to  correct  the  Influence  of  friction  by  any  graduation. 


14  LECTURE  I. 

Rightly  interpreted,  the  experiments  of  Him  furnished  results  which  you, 
no  doubt,  can  anticipate.  They  showed  that  the  steam  actually  gave  up  less 
heat  in  the  condenser  than  it  had  received  in  the  boiler,  and  that  the  heat 
absorbed  in  the  machine  was  proportional  to  the  actual  work  of  the  steam. 
The  ratio  of  these  two  quantities  furnished  a  new  determination  of  the  mechani- 
cal equivalent  of  heat,  which  nearly  approached  the  determinations  of  Joule 
and  Favre.  Thus,  although  individual  results  vary  within  considerable  limits, 
the  mean  of  Hirn's  experiments  gives  413,  or  just  the  same  as  that  found  by 
Favre  for  the  friction  of  steel  on  steel,  and  but  very  little  different  from  the 
results  of  Joule.  It  must  be  confessed  that  Hirn  drew  entirely  different  con- 
clusions from  his  experiments  ;  but  you  will,  I  think,  hardly  be  inclined  to 
coincide  with  his  views.  He  compared  the  heat  consumption  of  his  engine, 
not  with  the  entire  work  of  the  steam,  but  only  with  that  portion  correspond- 
ing to  its  expansion.  You  will  also,  I  think,  agree  with  me,  that  such  a 
division  into  two  parts  of  the  work  is  equivalent  to  the  assumption  that,  in 
the  period  preceding  expansion,  while  the  machine  works  with  full  steam 
pressure,  its  work  is  nothing,  and  that  appreciation  which  is  the  just  due  of 
the  skillful  experimenter  will  not,  I  trust,  blind  you  to  the  error  of  his  con- 
clusions.* (It  is  but  just  to  add  that  in  later  works  Hirn  has  acknowledged 
his  error  and  correctly  interpreted  his  experiments.) 


VI. 

You  will  now,  I  trust,  follow  with  confidence  the  generalizations  which  I 
shall  lay  before  you.  We  have  now,  in  fact,  arrived  at  the  same  results  by 
two  entirely  different  ways.  The  study  of  two  phenomena,  of  entirely  differ- 
ent character,  has  shown  us,  that  as  soon  as  all  the  heat  is  converted  into  work, 
in  both  cases,  we  obtain  the  same  numerical  relation  for  the  transformation. 
I  might  now,  without  trespassing  against  the  rules  of  experimental  methods, 
expect  you  to  recognize  in  this  a  perfectly  general  relation.  I  might  remind 
you  that  the  greatest  scientific  discoveries  are,  for  the  most  part,  the  result  of 
no  larger  number  of  experiments,  and  a  no  better  agreement  of  results.  I 
would  like,  however,  to  remove  every  lingering  trace  of  doubt,  and  prove  to 
you  that  it  is  impossible  for  two  different  experiments  to  give  as  a  value  for 
the  mechanical  equivalent  two  essentially  different  results,  i.  e.,  two  values, 
whose  difference,  if  any,  cannot  be  entirely  ascribed  to  unavoidable  errors  of 
observation. 

In  honor  of  Joule,  to  whom  we  owe  its  first  exact  determination,  let  us 
denote  the  mechanical  equivalent  by  J.  Let  this  be  the  value  as  determined 
by  observations  upon  the  steam  engine,  and  let  us  suppose  for  a  moment 
that  this  value  does  not  coincide  with  that  determined  in  some  other  manner. 
That  is,  suppose  that  we  are  able  by  the  expenditure  of  a  certain  amount  of 

work,  L,  to  generate  a  greater  amount  of  heat  than  y.     Let  this  amount  of 

heat  be  L  ,. 

J  (1  +  ^), 

'■'  See  Notes  S  aud  3  at  close  of  these  lectures,  also  Note  33. 


INTRODUCTION.  15 

and  let  us  suppose  tliat  it  is  then  applied  in  a  steam  engine  for  the  production 
of  work.     The  work  obtained  will  then  be 

X  (1  +  h), 

or,  what  is  the  same  thing,  we  have  stored  up  in  the  flv-wheel  of  the  engine  a 
living  force 

i  (1  +  h). 

This  living  force  we  can  now  convert  back  into  heat  by  the  first  process,  and 
we  thus  obtain  the  heat 

Again,  using  this  quantity  of  heat  in  the  steam  engine,  we  obtain  in  the 
fly-wheel  a  living  force 

X  (1  +  lif, 
and  accordingly  a  velocity  greater  than  before. 

But  the  steam  engine  and  the  other  apparatus,  whatever  it  may  be,  by 
which  the  work  is  transformed  into  heat,  may  be  considered  as  forming  one 
system.  It  follows,  therefore,  from  our  supposition,  that  in  any  period  during 
which  all  the  moving  parts  have  returned  to  their  initial  positions,  the  living 
force  has  increased  from 

L{\  +  Ji)      to      z  (1  +  ny. 

Perpetual  motion  is  therefore  accomplished.  The  supposition  is  therefore 
impossible. 

Inversely,  let  us  consider  a  process  by  which  heat  is  transformed  into  work, 
and  assume  that  it  is  possible  by  the  use  of  a  quantity  of  heat,  Q,  to  generate 
a  greater  amount  of  work  than  QJ.  The  consequence  of  this  supposition  will 
be  a  contradiction  similar  to  the  preceding.  For  this  process  it  may  be  re- 
marked that  the  steam  engine  is  a  reversible  apparatus.  Ordinarily  it  trans- 
forms heat  into  work,  but  by  the  application  of  outer  forces  its  actiou  may  be 
reversed,  and  work  may  be  transformed  into  heat.  The  motion  of  the  piston, 
caused  by  outer  forces,  will  gradually  vaporize  the  water  in  the  condenser, 
and  the  steam  thus  generated  will  be  compressed  in  the  cylinder  until  it  is 
changed  into  saturated  steam  of  the  temperature  of  the  boiler,  and  finally, 
this  steam  wll  be  compressed  into  water  of  the  boiler  temperature. 

The  steam  actually  brings  then  more  heat  to  the  boiler  than  it  starts  with 
in  the  condenser.  There  is  an  expenditure  of  work  and  a  generation  of  heat. 
In  order  to  obtain  perpetual  motion  nothing  more  is  necessary,  then,  than  to 
unite  in  one  system  a  steam  engine,  whose  actiou  is  reversed,  with  an  apparatus 
which,  according  to  the  supposition,  can  produce  from  the  quantity  of  heat  Q 
a  greater  work  than  QJ.  I  need  hardly  add  that  in  a  precisely  similar  way 
we  can  prove  that  no  process  can  give  for  the  mechanical  equivalent  any  other 
value  than  the  constant  one,  J.  Our  conclusions  have  thus  brought  us  to  a 
perfectly  general  natural  law.  Let  us  endeavor  to  formulate  these  conclusions 
into  a  series  of  principles  which  shall  accurately  express  their  essence  and 
make  evident  their  application. 

1.  To  "generate  heat  "  means  to  impart  to  the  ponderable  or  imponderable 
molecules  of  one  or  more  bodies  a  certain  amount  of  living  force  ;  if  the  bodies 


16  LECTURE  I. 

thereby  change  their  volume,  a  certain  work  is  performed  which  is  equivalent 
to  a  certain  quantity  of  living  force. 

2.  In  every  application  of  the  equation  of  work  it  is  necessary  to  take  into 
account  not  only  the  visible  living  forces,  but  also,  hy  means  of  the  mechanical 
equivalent,  the  heat  absorbed  or  set  free. 

3.  In  all  cases  in  which  we  fail  to  find  equilibrium  between  the  sum  of  the 
works  of  the  forces  and  the  change  in  the  living  forces,  or  when  such  equili- 
brium can  apparently  only  be  effected  by  the  introduction  of  an  empirical 
term,  as,  for  example,  by  introducing  the  work  of  friction,  or  by  the  assump- 
tion of  a  loss  of  living  force,  as  in  the  impact  of  bodies,  we  must  have,  to- 
gether with  the  mechanical,  heat  phenomena  also,  which  restore  the  equili- 
brium. 

4.  When  the  sum  of  the  works  of  the  forces  exceeds  the  increase  in  the 
sums  of  the  living  forces,  we  have  a  generation  of  heat  of  just  so  many  heat 
units  as  424  is  contained  in  such  excess.* 

5.  If  the  sum  of  the  work  of  the  moving  forces  is  less  than  the  increase  in 
the  sum  of  the  living  forces,  we  have  a  disappearance  of  heat  of  just  so  many 
heat  units  as  424  is  contained  in  such  difference. 

Is  it  necessary  to  insist  upon  the  importance  of  these  principles  ?  Who 
does  not  recognize  that  their  influence  extends  through  the  whole  range  of 
science '?  Who  can  fail  to  see  that  every  process  which  is  based  finally  upon 
motion  falls  under  the  scope  of  these  mechanical  laws  and  includes  in  it  an 
application  of  the  equation  of  living  forces?  Who  will  not  at  once  perceive 
that  every  application  in  which  these  laws  are  not  regarded,  must  be  at  once 
rejected,  as  soon  as  it  is  known,  or  even  suspected,  that  heat  phenomena  ai-e 
bound  up  with  the  mechanical  V  I  ventui-e  to  assert  that  there  is  not  a  natural 
science  which  can  elude  the  necessity  of  this  new  test.  Physiology  and  astron- 
omy have  need  of  it  equally  with  physics  and  chemistry. 

This  revision  of  scientific  results  is,  moreover,  not  merely  a  laborious  work 
of  correction,  which  at  most  allows  the  hope  of  discovering  in  certain  phenom- 
ena the  infiuence  of  certain  disturbing  causes  whose  effect  may  be  more  or  less 
difficult  of  calculation  ;  or  which  renders  more  exact  the  determination  of 
some  numerical  coefficient ;  it  constitutes  one  of  the  most  fruitful  studies 
which  true  science  can  undertake,  and  is  especially  suited  to  bring  to  light 
relations  between  apparently  the  most  diverse  phenomena.  The  single  exam- 
ple of  friction  teaches  us  what  tlie  new  theory  can  accomplish  in  directions 
which  are  generally  supposed  to  have  been  thoroughly  investigated  already. 


VII. 

Let  us  now  endeavor  to  test  the  value  of  these  considerations,  so  far  as  is 
possible  within  the  narrow  limits  to  which  we  must  confine  ourselves.  We 
shall  see  that,  from  the  very  first  step,  they  will  lead  us,  not  merely  to  super- 
ficial approximations,  but  to  exact  relations  which  may  be  verified  by  experi- 
ment. The  consequence  of  such  comparisons  must  be  a  continual  verification, 
d  posteriori,  of  the  absolute  generality  of  our  new  principles.     Let  us  first 

*  See  Note  4. 


INI  ROD  UCTION.  17 

consider,  as  is  most  natural,  the  changes  which  heat  causes  iu  the  volume  and 
condition  of  bodies. 

I  scarcely  need  remind  you  that  every  body  when  it  changes  in  tempera- 
ture changes  also  in  volume,  and  that  when  the  temperature  reaches  a  certain 
point  for  each  body,  that  sadden  change  takes  place  which  we  call  liquefaction 
or  linporization.  The  body  passes  from  the  solid  condition,  to  the  liquid,  or 
from  the  liquid  to  the  gaseous,  or  the  reverse.  No  part  of  science  has  been 
oftener  investigated,  and  yet,  in  the  absence  of  our  new  priuci[)les,  no  part 
seems  to  have  made  less  progress.  The  chapters  which  treat  of  this  subject, 
even  in  the  most  recent  text-books,  contain  little  more  than  a  presentation  of 
the  most  exact  experimental  methods  of  determining  the  most  reliable  coefB- 
cients  of  expansion,  the  specific  heat  and  latent  heat  of  substances,  and  tables 
in  which  these  numerical  values  are  given.  All  these  phenomena  are  given  as 
if  entirely  independent  of  each  other. 

This  want  of  connection  between  the  various  properties  of  the  same  body, 
or  between  similar  properties  of  different  bodies,  is  certainly  very  unsatisfac- 
tory. So  long  as  no  bond  of  union  exists  between  isolated  facts,  even  the  best 
observations  can  no  more  constitute  a  science  than  carefully  cut  stones,  ar- 
ranged in  order  of  size  and  shape,  can  constitute  a  building. 

It  is,  moreover,  worth  observing,  that  the  actual  progress  of  science  has,  at 
certain  periods,  rather  made  this  condition  of  things  worse  than  better.  The 
condition  in  physics  has  gradually  become  what  it  might  have  been  in  astron- 
omy, if  the  perfection  of  methods  of  observation  had  progressed  more  rapidly 
than  the  progress  in  theory — if,  for  example,  the  discovery  of  achromatism  or 
the  improvement  in  circle  graduation  of  recent  times  had  followed  imme- 
diately the  publication  of  Keppler's  laws,  instead  of  following,  as  they  did, 
long  after  the  discovery  of  the  universal  law  of  gravitation.  For  about  thirty 
years  science  possessed,  or  thought  that  it  possessed,  in  Mariotte's  laws,  the 
laws  of  the  expansion  of  gases,*  and  the  laws  of  Dulong,  Petit,  and  Neumann 
relating  to  specific  heat,  laws  analogous  to  those  of  Keppler.  The  marvelous 
improvement  in  experimental  methods  since  that  time,  recalled  by  the  mere 
mention  of  the  names  of  Rudberg,  Magnus,  and  Regnault,  led,  as  a  natural 
and  direct  consequence,  to  a  knowledge  of  the  deviations  of  these  laws  from 
the  reality,  and  there  were  no  theoretical  views  which  could  reconcile  these 
disagreements,  and  refer  both  laws  and  deviations  back  to  the  same  causes. 
The  importance  of  these  laws  themselves  soon  seemed  less  than  that  of  empiri- 
cal formulae,  which  represented  approximately  and  with  more  or  less  exact- 
ness the  general  features  of  the  phenomena.  Thus  it  was  that  science  seemed, 
little  by  little,  to  destroy  itself.  The  Mechanical  Theory  of  Heat  has  changed 
all  this.  It  has  not  only  put  a  new  phase  upon  the  phenomena  themselves,  but 
it  has  fundamentally  changed  our  conception  of  them  ;  in  many  cases  it  has 
even  pointed  out  the  reasons  of  variation.  If  we  assume  a  certain  amount  of 
heat  imparted  to  a  body,  the  volume  changes,  and  so  also  does  the  totality  of 
its  properties,  which  we  express  by  saying  that  its  "  temperature  is  increased." 
If,  however,  in  the  degree  that  a  body  is  heated,  we  increase  the  outer  press- 
ure upon  its  surface,  we  can  completely  prevent  its  expansion,  and  we  find 
that  in  this  case  the  amount  of  heat  necessary  to  raise  its  temperature  is  Icsa 

*  See  Note  5. 


18  LECTURE  I. 

than  before.  If  tlie  rise  of  temperature  is  in  both  cases  the  same  arbitrary 
imit  of  some  thermometric  scale,  then  the  two  quantities  of  heat  are,  the  one, 
the  specific  heat  by  constant  pressure,  the  other,  the  specific  heat  by  constant 
volume.  Their  difference  is  the  latent  heat  of  expansion.  The  expression 
"latent  heat"  means  simply  the  heat  imparted  to  the  body  which  has  no 
effect  upon  the  thermometer. 

What,  now,  is  the  mechanical  view  of  this  process?  To  heat  a  body — to 
draw  heat  from  a  certain  source  and  cause  it  to  enter  another  body — means  to 
diminish  the  living  force  of  the  source  by  a  certain  amount,  and  to  cause  in  the 
body  mechanical  processes  wliich  are  equivalent  to  this  diminution.  If  the 
volume  is  unchanged,  we  have  simply  an  increase  of  the  sum  of  the  living 
forces  of  the  particles  (rise  of  temperature),  and  it  may  be,  a  certain  work  due 
to  a  change  in  the  relative  position  of  the  molecules.*  If  the  pressure  is 
constant,  the  volume  increases,  and  there  is  a  neio  work  which  we  may  divide 
into  two  parts.  First,  the  distances  of  the  molecules  are  increased,  while  their 
mutual  actions  tend  to  keep  them  in  their  old  positions.  We  have,  therefore, 
a  work  performed  in  thus  separating  them,  whicb  we  may  call  "  disgregation 
work,"  and  regard  as  negative,  since  the  molecular  forces  oppose  the  displace- 
ments. Second,  the  body  expands  against  the  outer  pressure  of  the  atmos- 
phere. This  constitutes  another  work  which  is  also  negative,  and  which  we 
may  call  "outer work."  The  excess  of  the  specific  heat  by  constant  pressure 
over  the  specific  heat  for  constant  volume,  or  the  latent  heat  of  expansion,  is 
therefore  that  amount  of  heat  which  is  withdrawn  from  the  source  while  these 
works  are  performed.  Expressed  in  heat  units,  it  must  be  equal  to  the 
quotient  of  the  sum  of  both  these  works  divided  by  the  mechanical  equivalent. 

Consider  now  the  double  result  of  our  conclusions.  First,  we  have  learned 
what  latent  heat  is.  We  have  seen  that  it  is  that  heat  which  disappears 
when  work  is  performed,  and  which  reappears  again  when,  by  means  of  outer 
forces,  an  equal  work  of  opposite  sign  is  performed. 

In  the  second  place,  we  can  determine  a  numerical  relation  between  two 
physical  constants  which  are  apparently  independent  of  each  other,  and  also 
the  mechanical  work  corresponding  to  a  given  change. 

Unfortunately  this  relation,  in  the  form  in  which  it  occurs,  is  of  no  use. 
Of  the  two  terms  which  form  the  left  side  of  the  equation,  only  one,  that  which 
gives  the  "  outer  work,"  can  be  accurately  determined.  This  is  evidently 
equal  to  the  product  of  the  pressure  and  of  the  increase  of  volume,  and  is 
accordingly  quite  considerable  for  gases  and  vapors,  and  very  small  for  liquid 
and  solid  bodies.  The  disgregation  work,  on  the  other  hand,  in  the  present 
state  of  science,  eludes  every  attempt  at  determination,  and  will,  without 
doubt,  do  so  for  a  long  time  to  come.  We  must  have  a  complete  knowledge 
of  the  interior  constitution  of  the  body,  in  order  to  determine  it,  and  it  is 
impossible  to  say  how  far  the  more  or  less  plausible  ideas  held  to-day  repre- 
sent the  actual  state  of  things.  A  great  error  is  committed  if,  as  sometimes 
happens,  it  is  sought  to  establish  an  equivalent  relation  between  the  heat 
absorbed  by  a  body  and  the  outer  work.  The  error  may  be  diminished,  but 
not  eliminated,  by  replacing  the  disgregation  work  of  a  body  by  the  work  of 
outer  forces,  which  cause  a  deformation  equal  to  the  expansion.  It  cannot  but 
be  a  cause  of  wonder,  if  determinations  of  the  mechanical  equivalent  based 

*  See  Note  R. 


INTROn  UCTIO^\ 


19 


upon  such  a  mgtliod  of  determination,  have  given  results  which  closely  agree 
with  the  true  results.* 

In  view  of  these  diflBculties,  it  would  seem  as  if  the  theory  must  soon  cease 
developing,  and  as  if  the  discovery  of  exact  relations  whose  numerical  value 
can  be  checked  by  experiment,  must  be  delayed  till  such  a  time  as  the  science 
of  physics  shall  have  said  its  last  word  as  to  the  nature  of  all  things.  We 
can,  however,  avoid  such  difficulties  by  means  of  a  method  or  artifice  which 
we  owe  to  Sadi  Carnot.  Thus,  we  may,  without  knowing  anything  of  the 
interior  structure  of  bodies,  establish  such  relations  between  the  mechanical 
and  thermal  properties  of  bodies  as  shall  be  of  value,  by  considering  such  a 
sequence  of  changes  as  takes  place  during  a  process  in  which  the  initial  and 
final  conditions  are  alike,  and  hence  the  disgregation  work  zero. 

Let  us  consider  any  solid,  liquid,  or  gaseous  tody,  whicli  has  the  tempera- 
ture t,  the  pressure  p,  and  the  volume  v-  Let  us  call  the  state  of  the  body,  as 
determined  by  these  three  conditions,  the  state  T,  and  represent  the  volume  v  by 
the  abscissa  OA,  Fig.  2,  tlie  tension  p  by  the  ordinate  AT.  Now,  suppose  the 
outer  pressure  to  diminish,  and  while  the 
body  expands  let  us  impart  heat  to  it,  so 
that  its  temperature  changes  according  to 
any  given  law.  Let  this  continue  till  the 
body  comes  to  the  state  T' ,  for  which  it 
has  the  temperature  t' ,  the  volume  «',  and 
the  tension  p' .  Let  OB  =  v' ,  BT  =  j)  >  and 
let  the  abscissa  and  ordinate  of  the  curve 
TMT'  at  any  point  be  the  volume  and 
pressure  at  any  intermediate  state.  Call 
the  change  of  state  from  J"  to  T' ,  D. 
During  this  change  a  certain  amount  of 
heat,  Q,  is  imparted  to  the  body,  and 
a  certain  outer  work  is  performed,  L. 
Both  quantities  can  be  calculated,  if,  for  the  limits  of  temperature  t  and 
t' ,  the  influence  of  the  outer  pressure  upon  the  volume  of  the  body  and  the 
amount  of  heat  which  the  body  requires  for  a  given  change  in  volume  and 
temperature,  are  given  by  experiment.  These  quantities  may  be  expressed 
theoretically,  in  terms  of  the  coefficient  of  elasticity  and  the  two  specific  heats, 
provided  that  we  regard  these  two  elements  as  functions  of  the  temperature 
and  the  volume.  The  work  L  is  therefore  given  in  the  figure  by  the  area 
between  the  curve  TMT'  and  the  axis  OB  and  the  two  extreme  ordinates  AT 
and  BT'. 

Let  us  now  assume  that,  by  a  gradual  increase  of  the  outer  pressure,  the 
body  is  brought  back  to  its  original  state,  and  that  during  this  change,  which 
we  may  call  D' ,  we  continually  subtract  heat  from  the  body  as  it  is  compressed, 
so  that  its  temperatui'e  for  any  given  volume  is  less  than  during  the  change  I), 
except  at  the  beginning  and  end  of  the  entire  experiment.  The  body  thus 
finally  comes  back  to  its  original  condition,  but  at  all  intermediate  states  of  the 
change  D' ,  the  pressure  corresponding  to  a  given  volume  is  less  than  during 
the  change  B.  The  curve  TNT' ,  which  gives  this  second  relation  between 
pressure  and  volume,  consists  throughout,  with  the  exception  of  the  first  and 
*  See  Note  7. 


r 

r> 

\ 

P 

la' 

T* 

P^ 

V 

^ 

v. 

v'              I 

i 

Pig.  2. 


20  LECTURE  I. 

last  points,  of  less  ordinates  than  tlie  curve  TMT' .  The  area  between  the 
curve  TNT  ,  the  axis  OA,  and  the  extreme  ordinates,  gives  the  work  L'  per- 
formed upon  the  body  while  compressing  it,  and  evidently  we  must  have 

L  <L. 
We  may  also  compute  L'  and  Q'  in  the  same  way  as  L  and  Q. 

These  two  operations,  B  and  D  ,  may  be  regarded  as  parts  of  one  process  in 
which  the  initial  and  final  conditions  are  identical.  The  relative  position  of 
all  the  elements  of  the  body  are  the  same  at  the  beginning  and  end.  It  follows 
from  the  general  laws  of  mechanics,  that  there  must  be  a  complete  compensation 
between  the  work  of  the  molecular  forces  ;  that  the  inner  work,  corresponding 
to  the  transformation  D,  must  be  exactly  equal  and  opposite  to  that  which 
correspoiids  to  the  transformation  D' .  We  have  then  nothing  to  do  with  it. 
Still,  L'  is  less  than  L.  We  see,  therefore,  that  the  body,  in  the  cycle  of 
changes  to  which  it  is  subjected,  moves  in  a  determined  law  from  its  initial 
state  to  another,  and  then,  according  to  another  determinate  law,  returns  to  its 
original  state,  and  during  this  cycle  it  performs  an  outer  work  equal  to  4  —  4', 
which  is  represented  by  the  area  TMT'NC,  that  is,  by  the  difference  of  the  two 
areas  which  represent  the  works  L  and  L' .  No  inner  work  is  performed,  no 
sensible  living  force  has  disappeared,  therefore  a  certain  amount  of  heat  must 
have  disappeared  equivalent  to  the  work  done.  It  follows,  therefore,  first, 
that  the  body  during  the  change  D  has  received  more  heat  than  during  the 
change  D'  it  has  given  up.  Further,  the  ratio  of  the  work  L  —  L'  to  the  heat 
absorbed,  Q  —  Q',  is  equal  to  the  mechanical  equivalent.  The  formula 
L-L'=JiQ-Q'), 

which  we  thus  obtain,  gives  us  a  numerical  relation  between  the  mechanical 
and  thermal  phenomena,  the  study  of  which  is  usually  relegated  to  two  different 
departments  of  phy.sics,  since  L  and  L' ,  Q  and  Q' ,  are  detennined  by  means  of 
the  coefficients  of  elasticity,  the  two  kinds  of  specific  heat,  the  temperatures 
and  the  volumes.  We  may  obtain  as  many  special  relations  as  we  suppose 
cycles  of  changes.  In  order  to  obtain  a  general  equation  which  shall  include 
all  these  cases,  it  will  be  sufficient  to  consider  the  change  as  infinitely  small. 
Then  the  above  formula  will  become  a  differential  equation,  whose  integration 
will  give  us  the  law  of  expansion  of  the  body  under  all  circumstances.  Two 
other  differential  equations,  obtained  by  analogous  reasoning,  and  containing 
other  elements,  give  the  laws  for  melting  and  vaporization.* 

VIII. 

The  character  of  these  lectures  forbids  any  use  of  the  Calculus.  Without 
noticing  any  further,  therefore,  these  differential  equations  or  their  conse- 
quences, let  us  direct  our  attention  to  a  certain  class  of  bodies  of  which  we  can 
give  an  almost  complete  account,  simply  by  the  consideration  of  the  outer  work 
which  they  perform  under  the  action  of  heat.  It  has  for  a  long  time  been 
noticed  that  the  similarity  of  the  mechanical  and  thermal  peculiarities  of 
different  gases  seem  to  indicate  that  in  these  bodies  the  influence  of  the  mutual 
actions  of  the  molecules  is  not  noticeable. 

The  older  text-books  of  physics  held  generally  the  hypothesis  that  heat  was 

*  See  Notes  8  and  9. 


INTRODUCTION.  21 

something  material,  and  ascribed  the  elastic  force  of  gases  to  the  repulsive 
force  of  the  absorbed  heat  upon  the  molecules.  Laplace  himself  deduced  from 
such  views  the  law  of  Mariotte,  as  well  as  that  of  the  diffusion  of  gases  and 
of  their  expansion  {Mec.  Celeste,  liv.  sii.,  chap.  2).  At  the  present  time,  when 
the  views  as  to  the  uatujre  of  heat  have  undergone  such  great  changes,  the 
demonstration  of  Laplace  no  longer  holds  good ;  but  the  point  of  departure 
remains,  however,  the  same.  The  simplest  way  of  explaining  how  it  can  be 
possible  that  mechanical  action  and  heat  produce  almost  the  same  effects  upon 
various  gases,  is  to  assume  that  at  the  distances  which  separate  the  molecules 
of  such  bodies  their  mutual  actions  are  imperceptible.  The  laws  of  the  diffu- 
sion of  gases  seem  indeed  to  impart  to  this  conception  the  character  of  necessity. 
If  the  molecular  forces  in  gases  had  any  appreciable  intensity,  those  existing 
between  two  molecules  of  the  same  kind,  and  between  two  molecules  of 
different  kinds,  would  not  be  the  same.  The  properties  of  a  mixture  of  two 
gases  must,  then,  be  different  from  those  of  a  simple  gas.  Every  one  knows, 
for  example,  that  from  a  physical  standpoint  no  other  differences  exist  between 
oxygen  and  air  except  the  density  and  the  coefficient  of  refraction  ;  while  all 
those  properties  which  depend  upon  the  mutual  interaction  of  the  molecules 
are  exactly  the  same.  From  this  follow  two  consequences  :  First,  if  in  gases 
the  molecular  forces  are  almost  zero,  it  is  impossible  to  frame  any  conception 
of  the  constitution  and  general  properties  of  such  bodies,  without  assuming 
that  their  molecules  possess  a  considerable  velocity,  which  is  greater  the  higher 
the  temperature,  and  that  these  molecules  by  their  impact  cause  pressure. 
Second,  the  change  of  volume  of  a  gas  is  not  accompanied  by  any  disgregation 
work  at  all  comparable  with  the  outer. 

The  development  of  the  first  of  these  consequences  has  given  rise  to  views 
upon  the  constitution  of  gases  which  have  replaced  those  of  Laplace.  I  make 
here  merely  this  passing  reference,  as  I  do  not  wish  to  lay  down  anything  in 
these  lectures  which  rests  at  bottom  upon  hypothesis.*  The  second  conse- 
quence is  susceptible  of  direct  confirmation  by  experiment.  Thus,  if  we  allow 
a  gas  to  expand  without  overcoming  any  outer  resistance,  that  is,  without  per- 
forming any  outer  work,  and  if  the  disgregation  work  is  also  zero,  and  the  gas, 
both  at  the  beginning  and  end  of  the  experiment,  is  at  rest,  there  can  be  neither 
absorption  nor  generation  of  heat. 

This  assertion  may  excite  astonishment,  since  it  appears  at  variance  with 
well  known  facts.  All  of  us  are  familiar  with  the  simple  experiment  of  putting 
a  thermometer  under  the  receiver  of  an  air-pump,  in  order  to  observe  the 
decrease  of  temperature  which  occurs  at  the  very  first  stroke  of  the  pump. 
We  also  know  that  when  air  which  has  been  greatly  compressed  in  a  reservoir 
is  allowed  to  issue  in  a  jet  into  the  room,  it  may  cool  to  such  a  degree  that  the 
vapor  contained  in  it  is  frozen  and  deposited  upon  surrounding  bodies  in  the 
shape  of  frost. 

In  view  of  such  facts  it  appears  surprising  when  we  assert  that  a  gas  may, 
under  certain  conditions,  expand  without  cooling.  It  is  nevertheless  really  so. 
In  a  metallic  reservoir  B,  Fig.  3,  communicating  by  a  pipe  and  cock  with  the 
equal  reservoir  E,  Joule  has  compressed  air  under  a  pressure  of  22  atmos- 
pheres, while  the  reservoir  E  was  exhausted.     Both  were  then  immersed  in  a 

*  See  Note  10. 


22 


LECTURE  I. 


vessel  full  of  water  and  the  cock  I)  opened.  The  air  in  R  rushed  in  to  E,  and 
its  volume  was  thus  doubled,  while  there  was,  of  course,  no  resistance  to  its 
expansion,  except  the  small  amount  of  air  which  might  remain  in  E  after 
exhaustion.  Although  the  tension  of  the  air  decreased  from  22  to  11  atmos- 
pheres, there  could  be  then  no  outer  work  performed,  since  at  both  the  begin- 
ning .and  end  of  the  experiment  all  parts  of  the  apparatus  and  of  the  gas  were 
at  rest.  In  perfect  accord  with  theory,  it  was  found  that  there  was  no  absorp- 
tion of  heat.  The  most  sensitive  thermometer  immersed  in  the  water  which 
surrounded  both  R  and  E  showed  not  the  least  change  when  the  cock  D  was 
opened. 

It  is  not  difficult  to  see  why  under  the 
receiver  of  the  air-pump,  or  in  efflux  into 
the  air,  the  expansion  is  accompanied  by 
absorption  of  heat.  If  we  consider  closely 
the  case  of  the  air-pump,  we  see  that  a 
part  of  the  work  required  to  work  it  is 
furnished  by  the  pressure  of  the  air  re- 
moved. Outer  work  is  thus  performed  at 
every  stroke,  and  heat  correspondingly  ab- 
sorbed. We  could  not,  therefore,  have  a 
aiG.  a. 

better   confirmation  of   our  new   principles. 

In  the  efflux  of  air  it  rushes  with  great  velocity  from  the  reservoir,  driving  the 
outside  air  before  it  and  thus  performs  work.  Hence  the  cooling  with  which 
we  are  familiar. 

If  we  alter  Joule's  experiment  so  as  to  perform  outer  work  or  generate  living 
force,  we  shall  find  that  heat  is  absorbed.  Thus,  if  we  remove  the  reservoir 
E,  and  fasten  to  Z>  a  hose,  and  thus  allow  the  air  to  discharge  into  a  large  bell- 
glass  filled  with  water  and  inverted  in  the  pneumatic  bath,  the  water  will  be 
forced  out  against  the  pressure  of  the  atmosphere,  and  a  thermometer  in  R  will 
show  a  decrease  of  temperature  due  to  the  disappearance  of  heat  corresponding 
to  the  outer  work  performed.  We  can  easily  see  that  such  an  experiment  may 
lead  to  a  determination  of  the  mechanical  equivalent.  In  this  way  Joule  found 
441,  a  result  very  closely  agreeing  with  his  others,  the  deviation  being  com- 
pletely attributable  to  the  unavoidable  errors  of  observation.  Thus  disappears 
the  apparent  contradiction  between  what  we  may  call  the  old  and  new  physics. 

In  order,  however,  to  leave  not  the  slightest  doubt  or  uncertainty  upon  so 
important  a  point,  let  me  try  to  meet  in  advance  an  objection  which  has,  with- 
out doubt,  already  occuri'ed  to  you.  Let  us  look  somewhat  deeper  into  this 
process.  Conceive  in  the  reservoir  R  that  portion  of  the  gas  which,  after  the 
experiment  is  completed,  just  fills  this  reservoir.  Why  does  not  this  portion 
cool  during  the  expansion?  It  is  in  every  respect  similar,  and  in  similar  cir- 
cumstances, to  the  same  portion  of  the  gas  in  the  second  experiment,  where  its 
expansion  teas  accompanied  by  a  decrease  of  temperature.  In  both  cases  this 
portion  expands  against  the  pressure  of  the  rest  of  the  gas.  To  say  that  in  the 
one  case  it  preserved  its  temperature,  and  in  the  other  case  loses  it,  would  seem 
to  imply  that  it  knew  what  was  going  on  outside,  and  was  gifted  with  intelli- 
gence and  choice  of  action. 

We  do  not,  in  general,  willingly  receive  anything  against  a  theory  which  is 
regarded  by  the  highest  scientific  authorities  as  correct.     It  seems  a  thankless 


>  INTRODUCTION.  23 

task  to  give  audible  expression  to  difficulties  like  the  above  ;  yet,  at  the  bot- 
tom of  one's  heart  they  must  still  remain  and  cause  a  silent  distrust  of  all 
science.     Let  us,  therefore,  see  if  we  can  l&j  this  doubt. 

As  a  matter  of  fact,  that  portion  of  the  air,  in  Joule's  experiment,  which  re- 
mains behind  in  tlie  reservoir  B,  must,  and  does  lose  heat  and  cool,  because 
during  the  experiment  it  continually  imparts  living  force  to  that  portion  of  the 
air  which  rushes  with  considerable  velocity  into  the  reservoir  E.  But  this  liv- 
ing force  immediately  disappears.  The  velocity  of  the  gas  entering  E  is  de- 
stroyed by  the  friction  of  its  own  molecules  upon  each  other,  by  their  impact 
upon  the  walls  of  the  reservoir,  and  by  friction  in  the  communicating  pipe.  As 
soon  as  the  gas  ceases  to  enter,  therefore,  all  is  at  rest.  But  this  living  force 
cannot  be  destroyed  without  a  generation,  of  heat  exactly  equal  to  that  which 
disappears  in  the  reservoir  R.  In  Joule's  experiment,  then,  no  change  of  tem- 
perature was  observed,  because  there  is  a  perfect  compensation  ;  the  friction  in 
E  replaces  the  heat  disappearing  in  R.  We  have  no  need,  then,  to  ascribe  to  the 
gas  any  inconceivable  properties  ;  we  do  not  even  need  to  suppose  properties 
any  different  from  those  long  known.  We  can  also  easily  prove  our  conclusions 
by  experiment,  by  having  the  reservoirs  E  and  R  in  sexiarate  vessels,  when  R 
will  be  found  to  absorb  heat  and  E  to  give  out  a  precisely  equal  amount. 

This  remarkal)le  experiment  of  Joule,  performed  in  1845,  more  than  any 
other  directed  attention  to  the  new  theory.  Regnault  repeated  it  in  every 
shape,  with  all  the  precautious  which  his  long  experience  iu  calorimetric  re- 
searches rendered  available.  He  notified  the  Academy  iu  April,  1853,  that  he 
had  completely  confirmed  it,  and  from  that  moment  he  counted  himself  among 
the  advocates  of  the  new  views. 

No  further  doubt  can  remain.  In  gases  the  disgregation  work  which  accom- 
panies expansion  or  compression  is  zero,  or  at  least  is  imperceptible  to  ordinary 
calorimetric  methods.*  Heat  when  imparted  to  a  gas  causes  only  two  effects, 
a  rise  of  temperature  ("  vibration  work  ")  and  outer  work.  If  the  rise  of  tem- 
perature is  one  degree,  while  the  gas  expands  under  a  constant  pressure,  then 
the  outer  work  is  equal  to  the  product  of  this  pressure  into  the  increase  of 
volume.  If  F„  is  the  original  volume,  and  if  a  is  the  coeflficient  of  expansion, 
then  for  a  rise  of  one  degree  the  increase  of  volume  will  be  a  V„,  and  the  new 

V 
volume  will  be  F  =  F„  (1  -F  at)  for  a  rise  of  t  degrees,  or  V„  =  ^f^^f '  where 

t  is  the  temperature  for  the  volume  V. 

The  chanjre  of  volume,  then,  or  aY,,,  is  -^ ,  ,  and  this  multiplied  by  the 

^  ^        \  -\-  at 

pressure  p,  gives  us  for  the  outer  work 

^      1  +  at 
If  the  weight  of  the  expanding  gas  is  equal  to  one  unit,  then  the  value  of  the 
outer  work  is  the  mechanical  equivalent  of  the  excess  of  the  specific  heat  by 
constant  pressure  over  the  specific  heat  by  constant  volume.     If  J  represents 
the  mechanical  equivalent,  we  have 


24  LECTURE  I.  , 

or  if  V„  is  the  volume  for  tlie  temperature  zero  and  pressure  p^,  we  have 

{Gp-C,)J=ixi\V,. 
This  gives  us,  for  all  gases  which  follow  Mariotte's  law,  a  numerical  relation  be- 
tween the  coefficient  of  expansion,  the  two  specific  heats,  the  volume  of  the  unit 
of  weight  under  the  given  circumstances,  and  the  mechanical  equivalent  of 
heat.  We  may  make  use  of  it  in  order  to  determine  the  mechanical  equivalent 
by  means  of  the  physical  properties  of  various  gases,  and  since  for  most  gases 
these  properties  are  determined  with  a  degree  of  precision  which  cannot  at 
present  be  exceeded,  it  would  seem  that  in  this  way  we  should  obtain  a  value 
superior  to  all  the  others  in  accuracy.  The  formula  applied  to  air  gives  us 
the  number  436,  almost  identical  with  the  mean  of  Joule's  experiments,  if  we 
take  for  the  volume  of  the  unit  of  weight,  for  the  coefficient  of  expansion,  and 
for  specific  heat  for  constant  pressure,  the  values  given  by  Eegnault,  and  take 
for  the  specific  heat  for  constant  volume  the  best  value  as  given  by  experiments 
upon  the  velocity  of  sound.  The  agreement  of  this  calculation  with  the  ex- 
periments of  Joule  upon  friction  is,  in  fact,  most  remarkable. 

IX. 

Unfortunately  this  agreement  does  not  exist  when  we  apply  the  formula  to 
other  gases.  We  obtain,  however,  435 — a  very  close  value — for  hydrogen,  oxy- 
gen, and  nitrogen,  while  for  carbonic  acid  gas  we  obtain  a  value  considerably 
different.  Indeed  we  obtain  for  this  gas  two  very  different  values,  according  as 
we  take  one  or  the  other  of  the  two  determinations  of  Regnault  for  0'  and  100^.* 
For  other  gases  the  deviation  is  still  greater.  Whence  come  these  deviations  ? 
A  great  part,  without  doubt,  are  due  to  the  uncertainty  as  to  the  value  of  the 
specific  heat  for  constant  volume.  We  must,  however,  add  that  the  formula 
is  not  equally  reliable  for  all  gases,  since  the  disgregation  work  cannot  be  dis- 
regarded in  all. 

The  laws  of  Mariotte  and  Gay-Lussac  hold  accurately  for  no  gas  ;  they  are 
only  approximate  expressions  of  the  truth  for  those  gases  which  are  furthest 
from  their  points  of  liquefaction.  It  is  only  for  these  gases  that  the  agreement 
of  their  mechanical  and  thermal  properties  allows  us  to  assume  that  the  influence 
of  the  molecular  forces  is  zero.  On  the  other  hand,  gases  like  carbonic  acid, 
which  we  can  easily  liquefy,  whose  coefficient  of  expansion  is  five-tenths  greater 
than  air,  and  which  changes  very  rapidly  with  the  pressure  ;  gases,  finally, 
which  even  under  the  pressure  of  the  atmosphere  do  not  follow  the  law  of  Ma- 
riotte ;  for  all  such  we  have  every  reason  to  believe  that  a  noticeable  work  of 
the  molecular  forces  accompanies  changes  of  volume. 

If  we  apply  to  such  a  gas  a  formula  which  assumes  the  absence  of  all  dis- 
gregation work,  it  simply  shows  that  we  do  not  understand  the  principles  of 
which  we  make  use.  If  we  say,  as  has  been  said,  that  there  are  as  many  me- 
chanical equivalents  as  there  are  gases,  we  indirectly  declare  the  possibility  of 
perpetual  motion. 

It  would  seem  an  immediate  consequence  of  the  above,  that  by  repeating 
the  experiments  of  Joule  with  carbonic  acid  and  similar  gases,  and  determin- 
ing the  amount  of  heat  which  disappears  when  they  expand  without  perform- 

*  See  Note  12. 


INTRODUCTION.  25 

ing  outer  work,  we  inia:ht  obtain  a  measure  of  the  disgregation  work,  and  thus 
correct  the  above  formula  and  express  the  true  relations  between  the  different 
properties  of  such  gases.  Without,  however,  entirely  changing  the  experimen- 
tal methods  of  Joule,  there  seems  little  hope  of  obtaining  in  this  way  any  satis- 
factory results.  In  the  experiment  which  has  been  described,  the  expanding 
gas  is  surrounded  by  water,  and  even  under  a  pressure  of  23  atmospheres 
the  mass  of  the  gas  cannot  compare  with  that  of  the  water.  It  is  easy  to  com- 
prehend that  if,  for  example,  the  mass  of  the  water  is  only  twenty  times  that 
of  carbonic  acid  gas,  and  the  specific  heat  of  the  water  about  five  times  as 
great,  the  absorption  of  a  quantity  of  heat  which  would  change  the  temperature 
of  the  gas  one  degree  would  alter  that  of  the  apparatus  at  most  only  rnuth  of  a 
degree.  The  phenomena  in  question,  therefore,  would  be  completely  masked 
by  the  unavoidable  errors  of  experiment.  It  is  necessary  to  find  some  method 
of  doing  away  with  the  liquid  as  a  heat-measuring  substance,  and  observing  the 
change  of  temperature  in  a  stream  of  gas,  which,  without  performing  outer 
work,  experiences  a  considerable  change  in  elastic  force.  Under  such  circum- 
stances the  entire  heat  disappearing  is  equivalent  to  the  disgregation  work 
accompanying  the  expansion.  These  conditions  are  actually  complied  with  in 
an  experimental  method  conceived  by  Sir  William  Thompson.*  Our  space 
does  not  allow  us  to  describe  it  here.  The  application  of  this  method  to  hydro- 
gen, air,  and  carbonic  acid  has  shown  that  the  change  of  temperature  for 
hydrogen  is  almost  zero,  that  it  is  noticeable  for  air,  and  very  considerable  for 
carbonic  acid,  just  as  might  have  been  expected  from  the  experiments  of  Reg- 
uault.  Hydrogen  seems,  indeed,  the  furthest  removed  of  any  gas  from  its 
point  of  liquefaction.  Oxygen  and  nitrogen  show  a  less  perfect  accord  with 
the  properties  of  a  perfect  gas.     Carbonic  acid  gas,  finally,  deviates  decidedly. 

It  is,  therefore,  perfectly  natural  that  in  hydrogen  the  disgregation  work 
should  be  very  small — almost  zero— also  small  in  nitrogen  and  in  the  air, 
but  still  there — and  that  it  should  have  a  considerable  value  in  carbonic  acid. 
The  results  of  experiment  are  not  completely  satisfactory,  nor  exact  enough  to 
furnish  a  reliable  value  for  correction  of  our  formula.  They  suffice,  however, 
to  furnish  an  explanation  of  the  variation  which  has  been  found  in  the  mechani- 
cal equivalent,  as  determined  from  various  gases,  and  they  show  that  it  is  al- 
lowable to  use  the  formula  without  correction  for  air  and  hydrogen.  We  may 
consider  it  as  tolerably  certain  that  the  exact  value  of  J  lies  between  424  and 
426,  the  results  obtained  from  the  consideration  of  these  two  gases  ;  or  still 
more,  having  reference  to  the  uncertainty  in  the  value  of  the  specific  heat  by 
constant  volume,  between  420  and  430.  We  shall,  however,  continue,  in  what 
follows,  to  make  use  of  the  value  424. 

I  have  devoted  considerable  space  to  this  first  application  of  the  theory — 
much  more  than  I  can  give  to  others  of  which  I  intend  to  speak.  I  do  not  wish, 
however,  to  lay  any  especial  stress  upon  the  study  of  the  expansion  and  com- 
pression of  gases  ;  but  I  thought  it  well  to  show  thus  early  that  the  mechanical 
theory  of  heat  leads  to  results  which  agree  with  fact,  and  submits  to  calcula- 
tion not  only  known  phenomena,  but  also  predicts  new  ones,  and  that  this  pre- 
diction is  capable  of  numerical  verification. 

I  have  sought  to  excite  in  you  the  same  impression  which,  without  doubt, 


26  LECTURE  I. 

those  of  you  who  are  familiar  with  the  study  of  optics  have  already  experi- 
enced, as,  proceeding  from  the  uudulatory  theory,  they  made  their  first  applica- 
tions of  it  to  the  phenomena  of  reflection  and  refraction.  The  simplicity  with 
which  this  theory  harmonizes  known  facts,  the  fruitfulness  of  the  views  it  pre- 
sents, the  accuracy  of  its  predictions,  afford  the  most  convincing  proof  that  it 
closely  expresses  the  truth,  or,  at  least,  opens  a  path  which  leads  to  the  truth. 
I  shall  consider  my  ohject  obtained  if  this  first  lecture  shall  have  produced 
a  somewhat  similar  conviction. 


INTKODUCTION. 


SECOND    LECTURE. 


CONTENTS. 


I.  Eecapitalation  of  the  first  lecture.— Objects  of  the  second  lecture :  Investigation  of  heat 
engines  and  application  of  the  theory. 

II.  Comparison  of  steam  and  gas  engines.— Opposing  views  held  by  physicists  and  mechan- 
ics as  to  the  relative  value  of  these  two  engines. — Statement  of  tlie  considerations  by  which  the 
relative  disadvantages  of  the  steam  engine  liave  been  sought  to  be  proved.  These  considerations 
met  by  Hirn's  experiments. 

III.  General  expression  for  the  efficiency  of  a  Stirling  hot-air  engine.— This  expression  does 
not  indicate  any  superiority  over  the  steam  engine. 

IV.  Generalization  and  simplification  of  the  expression  for  the  efficiency  of  gas  engines. — 
Absolute  temperature,  absolute  zero  of  temperature.— Deduction  of  the  second  fundamental 
principle  of  the  mechanical  heat  theory  :  A  constant  ratio  exists  between  the  quantity  of  heat 
transformed  into  \\ork  in  a  perfect  heat  engine,  and  the  quantity  of  heat  transferred  from  a  hot 
to  a  colder  body.— One  real  advantage  of  the  gas  engine.— Practical  objections.— Advantages  of 
engines  witli  superheated  steam. — Engines  witli  two  fluids. 

V.  The  electro-magnetic  engine  may  be  regarded  as  an  heat  engine.— Experimental  proofs, 
by  Favre,  of  the  consumption  of  heat  in  this  engine  ;  another  determination  of  the  mechanical 
equivalent. 

VI.  The  necessity  of  induction  phenomena  shown  by  theory. 

VII.  Possibility  of  converting,  in  an  electro-magnetic  engine,  all  the  heat  into  work. — Why 
this  theoretical  advantage  is  not  practical. 

VIII.  Of  the  heat  generated  in  an  electro-magnetic  engine,  set  in  motion  by  outer  forces.— 
Determination  in  this  way  of  tire  mechanical  equivalent  of  heat  by  Joule.— The  experiment  of 
Foucault. 

IX.  Table  of  the  chief  determinations  of  the  mechanical  equivalent.— Remarks  upon.— Ap- 
plications of  the  new  theory  to  chemistry.— Measurement  of  the  worl-:  of  chemical  forces  by 
means  of  the  heat  generated. — Mechanical  explanation  of  some  electro-chemical  phenomena. — 
Mechanical  significance  of  the  measurement  of  electro-motive  force.— Experiments  of  Regnault 
upon  metal  amalgams. 

X.  Applications  to  animal  physiology. — Mayer's  theory  of  respiration  and  muscular  motion. 
—Experiments  of  Hirn  and  Beclard. 

XI.  Applications  to  botany.— Necessity  of  sunlight  for  vegetation.— Remarks  upon  the  com- 
mon origin  of  all  motion  on  the  earth. 

XII.  Upon  the  duration  of  the  sun"s  heat.— Hypothesis  of  Mayer.— Calculations  of  W. 
Thomson.— Remarlis  upon  tlie  scope  of  the  new  theory. 

XIII.  The  mechanical  theory  of  heat  reveals  laws  of  phenomena,  but  does  not  disclose  their 
mechanism. 

•  XIV.  Historical.— The  forerunners  of  the  theory  ;  Daniel  Bernoulli,  Lavoisier  and  Laplace, 
Rnmlbrd,  Davy,  Young.— Special  influence  of  Sadi  Carnot  and  Clapeyron.— The  discoverers  of 
the  new  principle  :  Mayer,  Colding,  Joule.— Helmholtz  and  his  treatise  upon  the  conservation  of 
force.— Works  of  Olausius.  Rankiue,  Thomson,  Zeuner. 

27 


LECTURE   II. 


We  have  now  passed  in  brief  review  the  phenomena,  by  the  consideration 
of  which  science  has  attained  to  the  recognition  of  the  new  principle  of  the 
equivalence  of  heat  and  work. 

Starting  from  the  laws  of  mechanics,  we  were  at  first  brought  face  to  face 
with  an  apparent  contradiction  between  these  laws  and  the  usual  theory  of  ma- 
chines. To  reconcile  this  contradiction,  it  was  necessary  to  include  the  phe- 
nomenon of  heat  among  the  mechanical  effects  which  occurred  in  the  entire  ma- 
chine during  its  motion.  The  heat  generated  by  friction  was  thus  found  to  be 
equivalent  to  the  difference  between  the  work  of  the  motive  forces  and  of  the 
resistances  ;  the  heat  absorbed  during  the  motion  of  the  machine  was  shown  to 
be  the  equivalent  of  the  work  done. 

The  correspondence  of  the  numerical  results  in  both  cases  gave  us  confi- 
dence in  the  correctness  of  our  views,  and  allowed  us  to  frame  a  very  precise 
idea  of  the  mechanical  equivalent  of  heat.  We  have  also  recognized  the  con- 
tradiction to  which  we  are  led  if  we  assume  that  the  value  of  this  equivalent 
can  'change  ;  and  we  have  still  further  convinced  ourselves  of  the  correctness 
of  our  new  principles  by  applying  them  in  various  ways.  Our  first  application 
was  with  reference  to  the  change  of  volume  or  of  condition  of  bodies  by  heat. 
For  solid  and  liquid  bodies  we  have  done  little  more  than  point  out  difficulties, 
and  briefly  notice  the  method  by  which  they  may  be  met.  We  have  treated 
gases  more  thoroughly.  Experiment  has  shown  us  that  for  air  and  other  gases 
far  removed  from  their  point  of  liquefaction,  the  disgregation  work,  or  the 
work  of  the  molecular  forces  which  accompanies  a  change  of  volume,  is  either 
zero  or  very  slight.  This  fact  has  permitted  us  to  com.pare  the  amount  of 
heat  which  must  be  imparted  to  a  gas  in  order  to  obtain  a  certain  amount  of 
outer  work,  with  that  work  itself.  Thus  we  found  a  new  determination  of  the 
mechanical  equivalent,  and  at  the  same  time  deduced  a  necessary  relation  be- 
tween the  different  mechanical  and  thermal  properties  of  the  same  gas. 

I  have  sought,  in  this  development,  to  keep  observation  and  theory  side  by 
side,  and,  in  some  degree,  show  you  that  every  experiment  was  the  realization 
of  an  idea  ;  and  finally,  to  make  it  as  evident  as  possible  how  firmly  all  parts 
of  our  new  theory  are  held  together. 

I  shall  now  pursue  an  opposite  method,  and  plunge  at  once  into  the  midst 

of  facts,  or,  so  to  speak,  into  the  midst  of  practical  industry,  and  I  shall  seek  to 

deduce  general  physical  laws  from  the  study  of  special  phenomena,  such  as  are 

presented  by  the  study  of  those  machines  which  derive  their  motive  power 

29 


30  LECTURE  11. 

from  the  action  of  heat.  The  investigation  of  "heat  engines  "  *  will  thus  form 
the  subject  proper  of  this  lecture.  The  lemaining  portion  of  It  will  seek  to 
give  you  a  review  of  those  applications  of  the  new  theory  which  lie  outside 
of  the  domain  of  physics,  and  especially  of  mechanics. 

II. 

There  are  two  kinds  of  heat  engines  of  special  importance  ;  the  steam 
engine  and  the  hot-air  or  caloric  engine. f  Upon  hot-air  engines  much  at  one 
time  was  said  and  written.  Great  stress  has  been  laid  upon  their  improve- 
ment, and  almost  unlimited  espectations  have  been  formed  of  their  mechanical 
efficiency. 

A  superficial  knowledge  of  Joule's  experiments  upon  gases  soon  spread, 
and  it  was  for  a  time  firmly  believed  that  soon  all  the  heat  furnished  by  the 
fuel  would  he  utilized.  On  the  other  hand,  after  Reguault's  experiments  upon 
the  latent  heat  of  vaporization,  which  seemed  to  show  that  only  an  inconsider- 
able fraction  of  the  power  of  the  heat  absorbed  could  be  utilized,  many  physi- 
cists formed  an  unfavorable  impression  of  the  steam  engine.  Thus  a  kind  of 
conflict  arose — I  can  hardly  say  between  theory  and  experience,  but  between 
one  view  which  apparently  harmonized  with  theory,  and  between  the  ever-ac- 
cumulating results  of  experience.  In  practice,  gas  engines  have  never  been 
found  of  such  economical  value  as  to  balance  the  difficulties  which  attend 
their  introduction.  Let  it  be  the  first  object  of  our  remarks,  therefore,  to 
point  out  the  significance  of  this  apparent  conflict.  For  the  sake  of  clearness, 
we  shall  take  a  numerical  example,  and  shall  therefore  choose  a  steam  engine 
working  under  a  steam  pressure  of  5  atmospheres,  or  with  a  steam  tempera- 
ture of  152°,  and  shall  first  assume  that  there  is  no  condenser.  The  steam 
enters  the  cylinder  in  the  saturated  condition  at  a  temperature  of  152"  ;  the 
formation  of  every  kilogram  of  steam  requires,  according  to  Regnault's  experi- 
ments, a  quantity  of  heat  denoted  by  the  number  653,  diminished  by  the  tem- 
perature, t,  of  the  feed  water.  As  it  enters  the  cylinder,  the  steam  raises  the 
piston  until  the  communication  with  the  boiler  is  closed,  when  it  expands  a 
certain  amount,  and  finally  discharges  into  the  air  under  the  pressure  of  the 
atmosphere.  If  we  assume  that  the  steam  remains  saturated  during  the  ex- 
pansion, then  the  final  temperature  is  100°,  and  every  kilogram  of  steam  which 
leaves  the  cylinder  takes  with  it  637  —  t  heat  vinits,  which  it  gives  up  in  con- 
densing into  water  of  the  temperature  i.  Out  of  the  653  —  t  heat  units  ab- 
sorbed in  the  generation  of  the  steam,  only  16  disappear  in  the  engine.  These 
16  units  are  transformed  into  work;  all  the  rest  is  wasted  in  the  atmosphere. 

Thus  if,  for  example,  t  is  only  10°,  only  -tr4-;jds,  or  less  than  .foth,  of  the 
heat  furnished  to  the  boiler  by  the  fuel  is  utilized.  This  fraction,  which  we 
may  call  the  "  efficiency,"  is  increased  somewhat  by  the  application  of  the  con- 
denser, but  it  is  always  very  small.  If,  for  example,  the  condenser  has  a  tem- 
perature of  40°,  and  the  steam  expands  in  the  cylinder  to  such  an  extent  that 

*  Under  heat  engines  we  include  all  those  machines  whose  motive  power  is  due  to  the  dis- 
appearance or  transformation  of  heat. 

t  Hot-air  engines  are  known  also  as  gas  engines  and  caloric  engines.  We  call  any  machine 
a  hot-air  engine  whose  action  depends  upon  the  heating  and  expansion  or  the  cooling  and  con- 
traction of  any  of  the  so-called  '■  permanent "  gases. 


INTRODUCTION.  31 

its  pressure  is  reduced  to  that  in  the  cnudenser,  ^vhich  never  can  be  the  case 
in  practice,  the  quantity  of  heat  which  one  kilogram  of  steam  brings  to  the 
condenser  is  only 

619  -  40  =  579  heat  units. 

If,  further,  the  condenser  is  fed  with  the  boiler  water  itself,  each  kilogram  of 
steam  requires  for  its  formation  only 

653  -  40  =  613  heat  units. 
Hence   34  heat   units   are   utilized,    and    the    efficiency   becomes    -J^nj-ths,    or 
about  pfth. 

The  condenser  is  therefore  of  considerable  advantage,  but  the  heat  utilized 
is  still  very  small  in  comparison  with  the  total  heat  imparted. 

These  are  nearly  the  words  of  Eegnault  in  his  criticism  of  the  steam  engine, 
which  has  been  extensively  repeated.  According  to  this,  the  most  important 
motor  of  our  civilization  is  but  a  very  imperfect  machine. 

Now  let  us  turn  to  experience.  The  treatise  of  Hirn,  from  which  we  have 
already  taken  several  important  results,  gives  us  the  data  for  a  reply.  We 
find  there  four  satisfactory  and  consistent  series  of  experiments  upon  the  steam 
engine,  which  are  nearly  identical  with  those  we  have  already  alluded  to.  The 
temperature  of  the  boiler,  as  a  mean  of  the  four  experiments,  was  146°,  that  of 
the  condenser,  34°.  Assuming  perfect  expansion,  we  have  for  the  efficiency,  in 
precisely  the  same  way  as  before,  -g=^,i7-ths,  which  is  nearer  to  -r^th  than  to  rgth. 
This  should  therefore  be  the  limit  which  can  never  be  exceeded,  and  which 
probably  experiments  can  never  show.  But  notwithstanding  this,  the  singular 
fact  remains  that  Hirn's  engines  gave  much  better  results.  The  excess  of  the 
heat  taken  by  the  steam  from  the  boiler,  over  that  given  out  in  the  condenser, 
that  is,  the  heat  expended  in  producing  work,  was  never  less  than  jTith  of  the 
total  amount  of  heat  ;  it  was  sometimes  even  ~th,  and  in  the  average  \i\\. 

Here,  then,  is  a  direct  contradiction.  Upon  the  one  hand,  a  theory  approved 
by  many  physicists  gives  for  the  efficiency  of  an  engine  a  value  but  little 
higher  than  -fpth  ;  upon  the  other  hand,  experiments  made  upon  machines  in 
actual  use,  which  therefore  must  he  very  far  from  perfect,  and  which  must  be 
fitted  with  special  apparatus  for  determining  the  efficiency,  give  a  result  twice 
as  great.  The  accuracy  of  the  experiments  is  proved  by  the  very  close  value 
of  the  mechanical  equivalent  which  they  have  given.  The  error  must  there- 
fore be  sought  in  the  theoretical  conclusions. 

Now  we  have  assumed  without  any  proof  that  the  steam  which  after 
expansion  leaves  the  cylinder  and  is  discharged  either  into  the  air  or  into  the 
condenser  is  saturated.  This  assumption  enabled  us  to  base  our  calculation 
upon  the  total  heat  of  vaporization  as  determined  by  Eegnault.  The  facts 
observed  by  Hirn  contradict  this  entirely  groundless  assumption,  and  prove 
that  the  phenomena  of  expansion  follow  much  more  complicated  laws,  and  that 
a  much  greater  part  of  the  heat  is  utilized.  The  steam  therefore  cannot  remain 
saturated  during  expansion.  Still  less  can  it  become  heated  above  the  point  of 
saturation,  and  become  at  the  end  of  expansion  superheated,  that  is,  possess  a 
less  elastic  force  than  that  which  corresponds  to  its  temperature  when  just 
saturated  ;  for  a  given  amount  of  superheated  steam  would  give  up  to  the  air 
or  to  the  condenser  more  heat  than  the  same  amount  of  saturated  steam,  and 
the  coefficient  of  efficiency  would  therefore  be  less  than  that  already  computed. 


32  LECTURE  IL 

There  is  only  a  tbird  supposition  possible,  viz. :  that  the  originally  saturated 
steam  condenses  during  its  expansion  in  the  cylinder,  and  a  part  of  it  becomes 
water.  This  supposition  is,  moreover,  correct.  We  can  point,  in  confirmation 
of  it,  to  an  almost  daily  occurrence  in  practice.  Every  one  knows  that  water 
collects  in  the  cylinder  if  it  is  not  jacketed.  Eankine  has  shown  that  the  prin- 
cipal cause  of  this  is  the  condensation  during  expansion,  and  not,  as  some  have 
thought,  the  accidental  introduction  of  water  from  the  boiler.  Hirn  has  given 
us  a  direct  experimental  proof.  A  copper  cylinder,  2  meters  long  and  0.15  in 
diameter,  was  closed  at  both  ends  by  thick  glass  plates.  Two  pipes,  with  cocks, 
were  connected,  the  one  with  the  boiler  and  the  other  with  the  air.  First  the 
air-cock  was  partly  opened  and  the  steam-cock  fully  opened.  Steam  thus 
entered  from  the  boiler,  drove  out  the  air  from  the  cylinder,  and  filled  it  with 
dry  and  perfectly  saturated  steam.  The  cylinder  was  then  as  transparent  as  if 
filled  only  with  air.  The  air-cock  was  now  fully  opened.  The  steam  escaped 
rapidly,  expanded  in  doing  so,  and  in  a  moment  the  cylinder,  before  so  trans- 
parent, became  perfectly  opaque,  and  the  condensation  was  visible. 

I  need  not  point  out  that  this  condensation  increases  the  amount  of  heat 
which  disappears  in  the  machine,  that  is,  which  is  turned  into  work.  Every 
kilogram  of  steam  which  reaches  the  cylinder  from  the  boiler  requires  for  its 
production  the  amount  of  heat  already  given.  But  the  heat  which  still  remains 
in  the  steam  when  it  enters  the  condenser  or  the  air,  is  diminished  by  the 
latent  heat  of  that  amount  of  steam  which  during  the  expansion  has  been 
liquefied. 

It  is  not  wholly  saturated  steam  which  leaves  the  cylinder,  but  a  mixture  of 
steam  and  water,  and  the  heat  converted  into  work  is  no  longer  equal  to  the 
difference  of  the  total  heat  of  vaporization  for  two  different  temperatures, 
but  is  equal  to  this  difference  increased  by  a  considerable  fraction  of  the  latent 
heat.  Condensation  during  expansion  is  thus  a  physical  property  of  steam,  to 
which  the  steam  engine  owes  a  large  part  of  its  efficiency.* 


III. 

Now  let  us  consider  the  hot-air  engine,  and  see,  if  we  can,  how  far  those 
hopes  are  justified  which  accompanied  its  discovery.  Without  doubt,  in  such 
an  engine  we  may  convert  all  the  heat  into  work,  when  our  object  is  sim- 
ply to  raise  a  loaded  piston  and  then  allow  it  to  sink  again.  Practice, however, 
demands  a  very  different  result ;  it  demands  a  continual  activity, — a  periodic 
motion,  which  shall  be  incessantly  repeated  so  long  as  heat  is  consumed.  For 
example,  it  is  required  that  the  piston  of  a  hot-air  engine,  after  it  has  been 
raised  to  a  certain  height,  shall  sink  again  to  its  original  position,  and  that 
this  action  shall  be  repeated  indefinitely.  But  the  air  under  the  piston  opposes 
the  downward  motion,  and  this  resistance  can  only  be  overcome  by  the  expen- 
diture of  a  certain  amount  of  work.  While  the  air  is  thus  compressed,  it 
becomes  heated,  and  this  heat  must  be  withdrawn  in  order  to  bring  about  the 
original  condition. 

If,  therefore,  during  the  first  period,  all  the   heat  imparted  can  be  con- 


*  See  Note  14. 


INTROD  UCTION. 


33 


verted  into  work,  in  the  second,  a  portion  of  the  work  thus  obtained  is  con- 
sumed in  order  to  generate  heat  ;  only  the  remainder  of  the  work  is  at  our 
disposal. 

The  question  is,  whether,  when  everything  is  taken  into  account,  the  hot- 
air  engine  possesses  any  peculiar  advantages.  Let  us,  as  an  examj^le,  take 
one  of  these  engines  whose  theory  is  simplest,  and  which  has  been  the  most 
tried  in  practice  ;  the  engine  of  Robert  Stirling,  which  dates  back  to  the  year 
1816. 

In  this  engine  the  air  is  first  heated  under  constant  volume,  then  it  expands 
under  constant  temperature,  it  is  then  cooled  to  its  original  temperature  while 
keeping  its  new  volume,  finally,  it  is  compressed  without  change  of  tempera- 
ture to  its  original  volume.  The  expansion  takes  place  under  a  much  higher 
temperature,  and  hence  under  a  much  higher  pressure,  than  the  compression. 
The  work  performed  in  the  first  case  is  greater  than  that  absorbed  in  the 
second,  and  the  excess  can  be  utilized. 

Let  us  represent  this  entire  cycle  of  changes  by  a  geometrical  construction. 
Let  OA,  Fig.  4,  be  the  volume  of  one  unit  in  weight  of  gas  for  the  initial  tem- 
perature ^„  and  let  the  ordinate  J.  T,,  be  the 
corresponding  pressure  ^)o-  The  air  is  first, 
without  change  of  volume,  raised  from  the 
temperature  to  to  the  temperature  ti,  which 
requires 

units  of  heat,  if  c,,  is  the  specific  heat  at  con- 
stant volume.  During  this  rise  of  tempera- 
lure,  the  pi'essure  rises  from  p,i  to^i,  repre- 
sented in  the  figure  by  AT^.  But  as  the 
volume  remains  unchanged,  no  work  is  j^er- 
formed.  The  pressure  upon  the  piston  sim- 
ply rises  from  p^  to  p^,  while  the  piston 
itself  does  not  move.  Now  the  load  on  the 
piston  is  gradually  diminished,  the  air  ex- 
pands, xcitliout  changing  in  temperature,  from  the  volume  «„  to  the  volume  «,., 
represented  by  OB.  The  temperature  remaining  constant,  the  volume  varies 
inversely  as  the  pressure,  and  the  curve  T-^  T, ,  which  is  approximately  an 
equilateral  hyi^erbola,  gives  the  law  of  change  of  volume  with  pressure ;  the 
last  ordinate,  BT^ ,  being  the  end  pressure.  Outer  work  is  performed,  which  is 
represented  by  the  area  AT^1\B.  But  while  the  air  is  thus  expanding,  in 
order  to  preserve  the  temperature  unchanged,  heat  must  be  added  to  the  air, 
which  heat  is  the  equivalent  of  the  outer  work  represented  by  the  area 
AT^T^B,  since  the  inner  work  in  case  of  air  is  zero.  In  the  third  opera- 
tion the  temperature  is  brought  back  to  ^O)  without  change  of  volume.  The 
pressure  accordingly  falls  from  BT^  to  BTq  without  any  expenditure  of  work, 
and  therefore 


1 

^1 

\ 

p. 

\ 

Ht 

To 

\ 

\ 

\ 

T, 

To 

1, 

"" 

To 

A'n 

J 

- 

I 

■v; 

I 

i 

C.{t^  -to) 


heat  units  must  be  abtracted,  if,  as  is  very  probable,  the  specific  heat  of  air 
for  constant  volume  is  independent  of  the  density.      In  the  fourth  and  last 


34  LECTURE  II. 

period  the  air  is  compressed,  while  its  temperature,  f„,  remains  constant, 
untilit  has  its  original  volume,  P^,.  Here  work  is  performed  upon  the  air,  and 
heat  must  be  abstracted  in  order  to  keep  the  temperature  constant.  The 
hyperbola  T^To  gives  the  relation  between  volume  and  pressure.  The  area 
ABToTo  represents  the  work  done  in  compression,  which  is  equivalent  to  the 
heat  q'  abstracted. 

The  air  receives,  therefore,  in  the  first  two  operations, 

c,{ti  —  fa)+q 
heat  units,  and  outer  work  is  performed  by  the  gas,  which  is  represented  by 
the  area  ATiTxB.     In  the  last  two  operations  heat  is  abstracted  from  the  gas 
equal  to 

dciti  —to)  +q' 

heat  units,  and  a  work  has  been  performed  upon  it  represented  by  the  area 
ABT^,T(,.  A  quantity  of  heat,  q  —  q  ,  has  thus  disappeared,  and  an  equivalent 
work  been  performed,  represented  by  the  area  T^T/f^  J",,.  The  heat  utilized 
is  g  —  q,  while  the  total  consumption  of  heat  is  d-.  {t^  —  to)  +  q,  and  the  heat 
not  utilized  at  all  would  seem  to  be  c,-  (t,  —  to)  +  q  ■ 

A  little  consideration,  however,  will  show  that  this  last  is  not  exactly  the 
case,  and  that  the  heat  not  utilized,  and  which  is  therefore  wasted,  is  q' .- 

The  quantity  of  heat  e„(#,  —  ^o)  which  the  air  loses  in  the  third  period, 
while  cooling  from  ty  to  ^o,  without  changing  in  volume,  can  be  utilized  in 
raising  the  temperature  of  another  portion  of  gas,  whose  weight  is  unity,  from 
to  toti.  This  portion  will  then  be  ready  to  expand,  performing  work  without 
change  of  temperature,  and  when  it  in  turn  is  cooled,  the  heat  given  up  by  it 
can  raise  the  first  portion  from  to  to  f ,,  and  so  on.  By  such  an  arrangement 
the  heat  c,:{ti  —  to)  may  pass  from  one  to  the  otlier  of  two  equal  portions  of  air 
which  keep  the  machine  in  motion.  Since  we  may  conceive  a  perfect  machine 
in  which  this  transfer  can  take  place  without  loss,  this  amount  of  heat  is  no 
part  of  the  useful  nor  useless  expenditure,  it  is  at  disposal  in  every  cycle.  It  is 
different  with  the  heat  q  which  the  air  parts  with  while  being  compressed  under 
constant  temperature.  Since  this  must  be  absorbed  by  a  cooling  apparatus, 
whose  temperature  is  t^,  it  cannot  again  be  used  in  order  to  raise  the  air  above 
this  temperature,  nor  to  keep  the  temperature  during  expansion  at  t^.  It  may, 
without  doubt,  be  used  in  another  machine  in  which  the  highest  temperature 
'does  not  exceed  ^o,  but,  so  far  as  the  first  is  concerned,  it  is  lost  entirely.  Hence 
we  have 

g-g' 
g 
•  as  the  ratio  of  the  heat  utilized  to  the  whole  heat  used. 

The  quantities  q  and  q'  are  easy  to  determine,  since  their  mechanical  equiv- 
alents are  represented  by  the  areas  AT-^T^B  and  ATqTqB.      Thus  if,  as 
■  always,  J"  is  the  mechanical  equivalent  of  heat, 
Jq  =  area  A  T,  T,  B, 
Jq'  =  area  AT,  ToB, 
q  —  q'_  area  AT,T,B  —  area  ATp  TpB 
q      ~  area.4r,r,5 

*  See  Note  15. 


INTBOD  UGTION.  35 

The  determination  of  the  hyperbolic  areas  is  best  performed  by  integration. 
We  have,  from  known  formulae, 

area  ATi  TxB  =p^Vo  log.  nat.  — , 
area  ATnT^B  =PqVo  log.  nat.  —  . 

But  ^1  and  po  are  the  pressures  of  the  same  mass  of  air,  at  the  same  volume, 
for  the  temperatures  ^,  and  ^o-  We  have,  therefore,  if  a  is  tlie  coefficient  of 
expansion, 

Px  _1  -^  ati 
Po~  1+  a  to  ' 
or,  finally, 

g-g'  ^  Ih  -Po  ^  a(ti~to) 
q  pi  1  +  ati  ' 

This  formula,  of  remarkable  simplicity,  gives  at  once  the  coefficient  of 
efficiency  of  an  engine  of  the  kind  discussed,  provided  that  we  only  know  the 
end  temperatures  between  which  the  engine  works.  It  is  also  evident  that 
this  coefficient  is  greater  than  is  ever  attainable  in  practice.  If,  now,  we 
assume  a  gas  engine,  working  between  the  same  limits  of  temperature  as  Hirn's 
engine,  we  have,  putting  ^^  =  146%  tg  =  34",  a  =  2T.id,  for  the  greatest 
efficiency  possible,  ilBths,  or  a  little  less  than  fths.  This  number  is  not  so  ■ 
much  greater  than  ^th,  which  Hirn  found  for  the  steam  engine,  that  one  can 
conclude  any  considerable  advantage  of  the  one  over  the  other.  It  would  not 
be  surprising  if  the  practical  imperfections  of  a  gas  engine,  working  between 
146'  and  34",  should  reduce  the  useful  work  to  ^th  of  the  total.  The  great 
importance  ascribed  at  one  time  to  gas  engines  does  not  appear,  therefore,  in 
any  degree  justified. 

IV. 

We  may,  however,  go  still  further,  and  prove  tliat,  considered  from  an 
economical  standpoint,  all  engines  which  work  between  the  same  temperatures 
give  the  same  result.  If  we  discuss  any  engine  in  the  same  way  as  we  have 
already  done  for  Stirling's,  we  sliall  see  : 

First,  that  in  any  given  engine  the  ratio  of  the  useful  to  the  total  expendi- 
ture of  heat  is  a  maximum,  when  no  heat  is  used  to  change  the  temperature  of 
the  gases,  or  when,  at  least,  this  heat  is  a  fixed  amount,  which  never  leaves 
the  system,  but  is  used  over  and  over  for  the  same  purpose. 

Second,  that  in  such  case  the  maximum  value  of  the  ratio  is 
a  it  I  -to) 
1  +  ati    ' 
where  t^  and  ^o  are  the  highest  and  lowest  temperatures*  which,  occur  in  the 
cycle  of  changes  in  the  machine.    We  may  give  to  this  expression  a  notewortliy 
significance   by  dividing   numerator   and   denominator  by   the  coefficient   of 
expansion,  a.     It  then  becomes 

t,   -to 

a 


36  LECTURE  11. 

for  whicli  we  may  write 


if  we  let 


T^  -n 


that  is,  if  we  count  the  temperature  from ,  or  from  — 373' C,  instead  of 

zero.  Now,  what  is  this  temperature,  which,  when  measured  from  —  273^  as 
the  zero  point,  thus  simplifies  the  expression  for  the  economical  coefficient? 
It  is  that  temperature  to  which  the  gas  is  raised  if  —  373"  is  the  zero  of  our 
thermometer,  and  if  the  gas  is  cooled-  under  constant  volume  to  —  273',  its 
pressure  would  be,  supposing  a  to  remain  constant,  zero.  This,  then,  is  the  tem- 
perature at  which  the  molecules,  while  preserving  their  original  distances 
apart,  which  they  possess  under  ordinary  temperatures,  become  motionless,  that 
is,  no  longer  impinge  upon  outer  bodies,  and  therefore  cease  to  exert  that  me- 
chanical effect  which  we  call  pressure.  In  a  word,  this  is  the  temperature  for 
which  the  sum  of  the  living  forces  is  zero.  But  the  expressions  living  force 
and  heat  are  identical,  and  we  can  thus  say  that  the  temperature  of  —  273°  C. 
is  the  absolute  zero  of  temperature. 

It  has  been  atteinpted  to  determine  this  point  in  various  ways,  and  at  one 
time  it  was  supposed  that  it  was  infinitely  removed  from  ordinary  tempera- 
tures. From  this  point  of  view  we  call  the  temperatures  T^  and  Tu  "absolute 
temperatures."  By  the  aid  of  this  definition  we  may  frame  the  following 
principle,  which  includes  the  theory  of  all  hot-air  engines  : 

"  In  every  hot-air  engine,  in  whatever  manner  it  works,  provided  only  no 
portion  of  the  heat  goes  to  useless  increase  of  temperature  of  the  air,  the  ratio 
of  the  useful  expenditure  of  heat  to  the  total  is  equal  to  the  difference  of  the 
absolute  temperatures  between  which  the  engine  works,  divided  by  the  greatest 
of  these  temperatures.'"* 

Does  not  the  simplicity  of  this  law  produce  the  impression  of  a  natural 

T,  —  Tn 
principle  ?    Does  it  not  appear  probable  that  the  expression  ~-^, indicates 

always  the  ratio  of  the  useful  to  the  total  expenditure  of  heat  in  a  heat  engine, 
whatever  may  be  the  changes  in  the  machine,  and  whatever  the  bodies  may  be 
which  are  made  use  of  for  the  transformation  of  the  heat  into  work  ? 

In  fact,  it  is  just  as  impossible  that  this  ratio  shall  have  two  different  values 
for  two  machines  working  between  the  same  limits  of  temperature,  as  that  the 
mechanical  equivalent  of  heat  should  have  two  different  values. 

Let  us  consider,  first,  that  in  a  heat  engine  the  excess  of  the  total  heat 
expenditure  over  the  useful  is  that  portion  of  the  heat  derived  from  the  fuel, 
which  during  the  action  of  the  machine  is  transferred  to  a  colder  body,  and 
thus  forever  lost  to  the  machine.  If  the  engine  is  reversible,  and  this  must 
necessarily  be  the  case  with  all  engines  worked  by  exp^msion  or  change  of 
condition,  it  will,  when  set  in  reverse  action  by  an  outer  force,  take  heat  from 
a  colder  body  and  transfer  it  to  a  hotter,  and  in  this  then  collects  all  the  heat 
generated  while  work  is  applied. 


INTRODUCTION.  37 

The  ratio  of  the  heat  thus  generated  to  the  total  quantity  of  heat  in  the 
hotter  body  is  exactly  equal  to  the  ratio  of  the  useful  consumption  to  the 
total  in  the  usual  action  of  the  machine. 

Let  us  now  assume  that  the  ratio  in  question  has,  in  two  different  machines, 
two  different  values.  It  is  not  diflBcult  to  conceive  these  two  machines  so 
united  that  the  one  in  which  this  ratio  has  the  greatest  value  shall  set  the 
other  in  reverse  action,  and  that  the  whole  work  developed  in  the  first  by  .the 
action  of  the  heat  is  completely  consumed  in  working  the  second.  The  action 
of  these  two  machines,  once  started,  would  continue  indefinitely  without 
expenditure  of  heat  or  of  work,  since  all  the  heat  consumed  in  the  first  machine 
would  reappear  in  the  second. 

Let  if  be  the  heat  consumed  in  the  one  and  reproduced  in  the  other  in  the 
same  time,  B  and  R"  the  ratios  of  useful  heat  consumed  to  total  in  the  first 
and  second,  and  let,  according  to  our  supposition, 
R  >  R". 

While  the  first  machine  consumes  the  heat  K  for  the  preservation  of  its 
motion,  it  transfers  from  a  source  of  heat,  whose  temperature  is  ^i,  a  quantity 

TT 

of  heat,  —  —  H,  to  the  condensing  apparatus,  whose  temperature  is  t(j.     In  the 

same  time  the  secoad  machine  reproduces  the  heat  H,  and  transfers  from  the 

condensing  apparatus  to  the  source  a  quantity  of  heat,  ^— ,  —  H.     The  final 

R 

result  of  the  combination  is,  that  from  a  cold  to  a  hotter  body  the  amount  of 

TT        TT 
heat,  ^  —  —  ,  is  transferred  without  the  expenditure  of  any  corresponding 

work.  If  this  result  is  not  a  contradiction  similar  to  the  production  of  perpetual 
motion,  it  is,  at  any  rate,  a  direct  contradiction  of  the  general  laws  which  we 
have  deduced  concerning  heat,  and  is  sufficient  to  show  that  the  hypothesis 
which  led  to  it  is  not  allowable. 

The  tendehcy  of  heat  to  pass  from  one  body  to  another  lies,  so  to  speak,  in 
the  definition  of  the  idea  of  "  unequal  temperature."  The  temperature  of  the 
body  which  gives  out  heat  is  the  highest,  and  that  of  the  body  which  receives 
it  is  the  lowest.  So  long  as  theory  cannot  define  with  sharpness  the  conditions 
which  we  denote  by  the  term  "  temperature,"  we  can  assign  no  decisive  reason 
why  there  must  be  but  one  sequence  of  temperatures,  and  we  may  be  inclined 
to  admit  that  it  is  not  impossible  that  bodies  which  do  not  interchange  heat, 
and  tlius  appear  to  have  equal  temperatures,  if  put  in  certain  mutual  relations, 
may  conduct  themselves  as  though  of  different  temperatures. 

All  experience,  however,  gives  a  most  decided  negative  to  this  supposition. 
It  shows  tliat  equality  or  inequality  of  temperature  is  an  absolute  fact,  inde- 
pendent of  the  experimental  process  by  which  it  is  made  evident.  If,  for 
example,  temperatures  are  recognized  as  equal  by  conduction,  they  are  also 
equa]  by  radiation.  It  is  not  possible  to  explain  this  law  by  even  the  most 
recent  advances  of  theory.*  It  is  sufficient  that  all  the  facts  justify  us  in 
recognizing  in  this  an  absolute  principle. 

Fourier  has  based  upon  this  principle  his  theory  of  radiant  heat  and  of 

*  See  Note  18. 


38  LECTURE  11. 

equilibrium  of  temperature,  and  altliouarli  this  tlieory  may  seem  unsatisfactory 
in  view  of  the  discovery  of  the  diversity  of  heat  rays  and  of  elective  al)sorption, 
as  shown  by  most  bodies,  and  may  have  therefore  seemed  unsatisfactory  to 
many,  still  all  doubt  must  vanish  in  view  of  the  wonderful  discoveries  tc 
which,  by  a  new  application  of  the  same  principle,  KirchhofE  has  been  con- 
ducted. 

We  assume,  however,  nothing  hypothetical,  and  we  base  ourselves  upon 
the  most  certain  facts  of  experience  when  we  propound  the  following  prin- 
ciples : 

First,  it  is  impossible  for  heat  to  pass  from  a  cold  to  a  warmer  body  without 
being,  at  the  same  time,  accompanied  by  some  phenomena  which  may  be 
regarded  as  the  cause  of  such  transfer.  Especially  in  no  machine  in  which 
neither  heat  nor  work  is  given  out  can  any  such  transfer  occur. 

Second,  it  follows  necessarily  from  this  first  law  that  the  ratio  of  the  useful 
to  the  total  expenditure  of  heat,  in  a  machine  whose  action  depends  upon 
change  of  volume  or  of  condition  of  aggregation,  is  independent  of  the  consti- 
tution of  the  body,  and  is  determined  solely  by  the  extremes  of  temperature 
between  which  the  machine  works,  provided  that  heat  is  not  consumed  in 
bringing  about  change  of  temperature.     The  formula 

T,  -  n 


which  we  have  found  directly  for  the  hot-air  engine,  holds  good  therefore  for 
every  engine.  It  shows  us  at  once  that  if  any  engine  is  superior  economically 
to  another,  it  is  not  because  the  bodies  which  serve  to  transfer  heat  and  con- 
vert it  into  work  possess  this  or  that  property.  The  only  advantage  which 
one  body  can  present  over  another  is  in  the  wider  limits  of  temperature  ren- 
dered available. 

From  this  point  of  view  the  superiority  of  thehct-air  engine  over  the  steam 
engine  becomes  evident.  We  cannot  have  the  temperature  of  a  steam  boiler 
much  above  150^  or  160°,  because  the  very  rapid  increase  of  pressure  for  higher 
temperatures  would  require  extraordinary  thickness.  Since,  on  the  other 
hand,  it  requires  but  little  less  than  a  rise  of  temperature  of  273''  to  increase 
the  pressure  of  a  gas  one  atmosphere,  we  see  what  enormous  temperature 
limits  we  may  ha.ve  in  a  hot-air  engine,  without  greater  strength  than  that  re- 
quired for  an  ordinary  high-pressure  steam  engine. 

W^e  should  have,  then,  greater  economical  advantages,  were  we  not  opposed 
by  practical  difficulties  ;  as,  for  instance,  the  oxidation  and  the  rapid  deteriora- 
tion of  metal  which  always  accompanies  highly  heated  air.  The  use  of  super- 
heated steam  would  seem  to  remove  this  objection,  without  greatly  diminish- 
ing the  peculiar  advantages  of  the  gas  engine.  Superheated  steam  is,  in  fact, 
a  gas  ;  its  pressure  near  the  point  of  saturation  increases  undoubtedly  with  ris- 
ing temperature  more  rapidly  than  that  of  air  ;  but  all  its  thermal  and  mechan- 
ical properties  coincide  more  with  those  of  air  the  higher  its  temperature  is 
raised. 

Future  progress  would  therefore  seem  in  the  direction  of  such  an  applica- 
tion, in  which  the  peculiar  advantages  of  the  hot-air  engine  are  combined  with 
those  of  the  steam  engine. 

Engines  working  with  two  kinds  of  steam,  of  which  much  has  been  heard, 


INTRODUCTION.  39 

are  an  attempt  to  increase  tlie  mechanical  efficiency  of  the  steam  engine  by 
diminishing  the  lower  limit  of  temperature.  The  water-steam  which  is  con- 
densed in  the  condenser  is  made  to  heat  and  vaporize  a  more  volatile  liquid, 
like  ether  or  chloroform.  This  new  steam  works  a  second  engine.  It  thus 
becomes  possible  to  lower  the  temperature  of  the  condenser  below  that  which 
would  exist  for  water-steam  alone.  The  increase  of  motive  power  in  such  a 
construction  is  shown  by  the  diminution  of  T,,  in  our  formula  ;  but  it  is  evident 
that  this  gain  is  not  comparable  with  that  which  we  may  obtain  in  the  steam 
engine  with  superheated  steam,  by  increase  of  the  upper  limit  T, . 

V. 

There  remains  still  a  third  kind  of  apparatus  which  we  may  include  among 
heat  engines,  although  it  is  apparently  totally  different  from  the  hot-air  and 
steam  engine,  viz.,  tlie  electro-magnetic  engine  ;  and  in  spite  ©f  the  small 
practical  results  thus  far  attained  by  it,  these  lectures  would  be  incomplete 
without  some  discussion  of  its  value  and  efficiency. 

There  is,  moreover,  from  a  purely  scientific  standpoint,  hardly  a  subject 
more  fruitful  in  interesting  and  novel  views  than  the  theory  of  the  electro- 
magnetic engine,  and  I  shall  therefore  devote  at  least  as  much  space  to  it  as 
to  the  comparison  between  the  steam  and  hot-air  engine,  even  although  its 
practical  value  at  present  is  less. 

If  we  neglect  differences  of  detail,  we  may  divide  all  electro  magnetic  en- 
gines into  two  classes  ;  either  oscillating  or  rotary.  In  the  oscillating  engine  a 
fixed  wire  coil  or  an  electro-magnet,  as  soon  as  the  current  passes,  attracts 
another  wire  coil,  electro-magnet,  magnetized  steel  rod,  or  a  piece  of  soft  iron. 
As  soon  as  this  movable  piece  comes  in  contact  with  the  fixed,  by  means  of  a 
circuit-breaker,  the  attraction  is  changed  into  repulsion,  or  it  is  neutralized  by 
the  attraction  of  another  piece.  The  motion  is  thus  reversed,  and  this  action 
is  repeated  indefinitely.  Such  a  motion  can  be  utilized  in  a  manner  similar  to 
that  of  a  piston. 

In  the  rotary  engine  the  fixed  and  moving  pieces  are  situated  in  the  radii 
of  two  concentric  circles  or  wheels.  When  the  current  passes,  the  movable 
wheel  strives  to  take  its  position  of  stable  equilibrium,  but  in  the  moment  at 
which  this  is  reached  the  circuit-breaker  acts,  the  wheel  is  carried  round  by 
its  momentum,  and  a  continuous  rotary  motion  is  the  result.  This  motion 
can  be  utilized  like  any  other  motion  of  the  same  kind  produced  by  any 
mechanical  force.  In  both  cases  the  principle  of  construction  and  the  origin 
of  the  force  is  the  same.  The  action  of  the  currents  or  of  the  magnets  strives 
to  bring  about  a  condition  of  stable  equilibrium,  and  some  physical  change  in 
the  system  at  the  moment  that  this  condition  is  satisfied  continues  the  motion. 
What  is  the  mechanical  expression  for  this  entire  action'? 

Let  us  first  consider  the  case  in  which  the  machine,  in  spite  of  the  passage 
of  the  current,  is  held  fast  without  motion.  The  electromotive  voltaic  chain, 
or  battery,  and  the  engine,  form,  then,  a  fixed  system,  in  which  two  kinds  of 
processes  are  simultaneously  going  on.  In  the  battery  there  is  in  a  given  time 
a  certain  amount  of  chemical  action  ;  at  the  same  time,  in  all  the  conductors 
through  which  the  current  passes,  heat  is  generated,  and,  so  long  as  the  ma- 
chine is  not  in  motion,  this  is  all.     In  the  battery  we  have  chemical  action  ; 


40  LECTURE  II. 

atoms  obey  their  affinities,  pass  from  one  condition  to  another  in  which  their 
affinities  are  satisfied,  and  so  into  equilibrium.  From  the  definition  of  mechan- 
ical work,  it  follows  that,  in  such  a  series  of  changes,  positive  work  is  per- 
formed, lu  the  system  of  conductors  which  the  stream  passes  through,  there 
is  generated  a  certain  amount  of  living  force  in  the  shape  of  heat.  Necessarily 
there  must  be  an  equivalence  between  the  work  of  the  chemical  forces  and  the 
heat  simultaneously  developed  in  the  conductors  and  battery.  A  given  amount 
of  chemical  action  of  a  given  kind  must  correspond  to  the  generation  of  a  con- 
stant amount  of  heat,  whatever  may  be  the  constitution  of  the  circuit  and  of 
the  battery. 

These  theoretical  conclusions  are  confirmed  by  a  remarkable  experiment  by 
Joule  and  Favre.  A  large  calorimeter,  which  was  essentially  simply  an  im- 
mense mercury  thermometer,  with  two  cavities  in  its  bulb  for  the  reception  of 
bodies,  was  used  by  Favre,  and  the  following  determinations  made  :  First,  in 
one  of  the  cavities  was  placed  a  simple  galvanic  element  of  zinc  and  platinum 
immersed  in  acid  water,  united  by  a  very  short  copper  wire.  Thus  was  deter- 
mined the  amount  of  heat  generated  by  one  equivalent  of  zinc  when  decom- 
posed, taking  the  equivalent  at  66  grams.  The  mean  of  a  number  of  good  de- 
terminations showed  that  this  amount  of  heat  was  sufficient  to  raise  37,360 
grams  of  water  one  degree.  Then  the  thick,  short  copper  wire  was  replaced  by 
a  thin  wire  of  considerable  length,  wound  in  a  spiral.  The  decomposition  of 
a  given  quantity  of  zinc  then  was  found  to  give  a  much  less  quantity  of  heat, 
and  the  diminution  was  greater  the  longer  and  thinner  the  wire.  The  wire 
itself  was  notably  heated,  and  when  it  was  included  in  the  other  cavity,  so  that 
the  total  heat,  both  in  the  elements  and  circuit  was  determined,  the  sum  was 
found  to  be  precisely  the  same  as  in  the  first  experiment.  The  decomposition 
of  66  grams  of  zinc,  again,  generated  37,360  heat  units.  Eepeated  in  the  most 
diverse  ways,  with  conductors  and  elements  of  the  most  diverse  kind,  the  same 
results  were  always  obtained  :  so  that  in  all  cases  in  which  the  action  of  the 
current  performed  no  outer  work,  the  heat  in  the  entire  circuit  and  the  chemical 
action  were  found  to  be  perfectly  equivalent. 

If  now  the  machine  is  in  motion,  living  force  is  generated  or  work  is  per- 
formed outside  of  the  circle,  as,  for  example,  the  raising  of  a  weight  to  a  cer- 
tain height.  If  the  heat  generated  in  the  circuit  should  still  remain  the  same, 
we  should  have  at  one  time  the  work  of  the  chemical  forces  in  the  battery 
equivalent  to  a  certain  quantity  of  heat,  and,  at  another  time,  equivalent  to  the 
same  heat  increased  by  a  certain  mechanical  work,  which  is  plainly  impossible. 
Accordingly,  if  by  the  action  of  the  current  in  any  system  of  spirals  or  electro- 
magnets, outer  work  is  performed,  there  must  be  a  diminution  of  the  heat  gen- 
erated in  the  entire  circuit  by  a  given  amount  of  chemical  action,  and  this 
diminution  must  be  the  exact  equivalent  of  the  outer  work  performed.  Experi- 
ment has  confirmed  this  conclusion.  In  the  second  cavity  of  his  calorimeter 
Favre  replaced  the  conducting  wire  of  the  previous  experiment,  by  a  very  small 
electrormagnetic  engine,  which,  by  means  of  a  mechanism  unnecessary  here  to 
describe,  raised  a  weight.  Under  these  new  conditions  the  decomposition  of 
66  grams  of  zinc  generated  less  than  37,360  heat  units,  and  the  observed  differ- 
ence stood  in  a  constant  relation  to  the  work  of  the  engine.* 

r  See  Note  19. 


INTRODUCTION.  41 

Every  unit  of  heat  was  tlius  found  to  correspond  to  rai  outer  work  of  443 
units.  The  difference  between  this  number  and  the  mechanical  equivalent  of 
heat,  as  determined  by  Joule,  or  as  determined  from  the  properties  of  gases, 
does  not  exceed  the  limits  which  may  properly  be  ascribed  to  errors  of  obser- 


VI. 

In  an  electro-magnetic  engine  there  is,  therefore,  a  loss  of  heat  as  soon  as 
mechanical  work  is  performed,  and  hence  it  is  with  perfect  propriety  that  we 
have  classed  such  machines  among  heat  engines.  The  mechanical  power  is 
due  to  a  partial  transformation  of  the  heat  caused  by  chemical  action  in  the 
battery  ;  just  as,  in  the  steam  engine,  it  is  due  to  a  partial  transformation  of 
the  heat  caused  by  combustion  of  fuel  under  the  boiler.  In  the  one  as  in  the 
other  case,  such  transformation  depends  upon  certain  physical  laws,  which 
may  be  regarded  as  so  many  general  consequences  of  the  mechanical  theory  of 
heat.  The  study  of  the  steam  engine  has  revealed  to  us  the  condensation  of 
steam  when  expanding  ;  the  study  of  the  electro-magnetic  engine  makes  evi- 
dent to  us  the  necessity  of  the  phenomena  of  induction. 

There  is  but  one  way  in  which  we  can  comprehend  how  the  motion  of  a 
machine  can  diminish  the  amount  of  heat  generated  in  a  conducting  wire  by  a 
certain  amount  of  chemical  action.  The  generation  of  heat  in  a  unit  of  time 
is  proportional  to  the  square  of  the  intensity  of  the  current,  while  the  intensity 
itself  is  proportional  to  the  amount  of  chemical  action  in  the  same  time.  It  is 
evident  that  the  heat  generated  by  the  decomposition  of  one  equivalent  of 
metal  is  directly  proportional  to  the  intensity  of  the  current  which  causes  this 
chemical  action,  or  inversely  proportional  to  the  time  of  decomposition.  If, 
thus,  i  is  the  intensity  of  current,  and  t  the  number  of  seconds  required  for  the 
decomposition  of  one  equivalent  of  metal,  then,  since  the  chemical  action  and 
intensity  of  current  are  proportional,  the  product  i  t  is  equal  to  a  constant,  Tc. 
The  heat  generated  by  the  decomposition  of  one  equivalent  of  zinc  is  propor- 

tional  to  Pt,  or  can  be  represented  by  iJc,  or  by  — -.     It  is  therefore  necessary 

that  the  chemical  action  in  a  battery,  the  current  from  which  Vv'orks  an  electro- 
magnetic engine,  must  be  lessened,  and  hence  the  intensity  of  the  current 
diminished,  by  the  motion  of  the  engine.  If  a  galvanometer  is  interposed  in 
the  current,  its  deviation  during  motion  of  the  engine  must  be  less  than  when 
the  engine  is  at  rest,  and  the  difference  will  be  greater,  the  greater  the  work  of 
the  machine  corresponding  to  a  given  chemical  action.  This  is  completely  con- 
firmed by  experiment.  There  can  be  no  doubt  as  to  the  fundamental  fact  that 
the  motion  of  an  electro-magnetic  engine  diminishes  the  intensity  of  the  cur- 
rent. What  can  be  the  cause  of  this  diminutioii?  Is  it  an  increase  of  the 
resistance  to  the  current?  Is  it  a  process  similar  to  that  which  separates  and 
puts  in  motion  the  two  kinds  of  electricity  in  the  battery  ? 

An  increase  of  resistance  is  impossible,  because  experiment  has  proved  that 
the  resistance  of  a  conductor  is  the  same,  whether  it  is  at  rest  or  in  motion.  It  is 
therefore  necessary  that,  in  a  machine  whose-parts  are  relatively  in  motion,  a  cur- 
rent shall  tend  to  give  rise  to  an  opposite  current,  or,  using  the  customary  expres- 
sion, an  electro-motive  force  shall  be  generated,  opposed  to  that  of  the  battery. 


42  LECTURE  II. 

But  whatever  takes  place  in  a  machine  in  consequence  of  its  motion,  must 
take  place  also  in  any  system  of  conductors  and  currents  when  in  any  sort  of 
motion.  If,  therefore,  a  closed  conductor  is  moved  in  the  neighborhood  of  a 
magnet  or  current,  the  motion  must  cause  in  the  conductor  a  current  o])posed 
to  that  which  would  have  to  pass  through  it,  in  order,  by  means  of  electro- 
magnetic forces,  to  continue  such  motion. 

In  this  sentence  you  have  already  recognized  one  of  the  fundamental  laws 
of  induction,  and  it  would  not  be  difficult  to  prove,  analogically  at  least,  in 
similar  manner,  all  those  discoveries  which  have  rendered  so  famous  the  name 
of  Faraday.  Induction  currents,  the  existence  of  which  seemed  at  first  so  mar- 
velous to  physicists,  were  observed  by  Ampere  *  ten  years  before  Faraday,  but 
without  his  daring  to  believe  them.  It  was  sought  in  vain  to  deduce  them 
from  the  phenomena  of  static  electricity.  By  means  of  the  mechanical  theory 
of  heat  they  receive  their  true  interpretation.  The  generation  of  induction 
currents  are  the  means  which  nature  employs  in  order  to  produce  work  in  the 
electro-magnetic  engine.  The  laws  governing  induction  currents  are  such  that 
the  equation  of  living  forces  is  fulfilled  both  for  motion  of  the  machine  as  well 
as  for  rest. 

If  we  consider  on  one  hand  the  known  expression  for  the  mutual  action  of 
two  currents,  and  on  the  other  the  proportionality  of  the  heat  generated  by  the 
current  to  the  square  of  the  intensity,  as  given  by  experiment,  and  unite  these 
two  facts  by  the  principle  which  we  have  deduced,  we  can  determine  generally 
both  the  direction  and  intensity  of  the  induced  current  generated  by  the  relative 
motion  of  a  current  and  a  closed  conductor.  We  may  in  this  way  discover  all 
the  laws  deduced  by  Neumann,  in  1845,  in  an  entirely  different  manner,  upon 
which  chiefly  rests  his  scientific  fame. 

This  remarkable  relation  between  the  mechanical  theory  of  heat  and  the 
phenomena  of  induction  was  first  made  known  by  Helmholtz,  in  1847. 


VII. 

Among  the  number  of  laws  deduced  in  this  manner  by  theory,  and  con- 
firmed by  experiment,  is  that  of  the  proportionality  of  the  induced  current  to 
the  velocity  of  the  motion  producing  it.f  As  the  motion  of  an  electro-magnet 
is  accelerated,  the  electro-motive  force  of  induction  increases,  and  hence  the 
current  intensity  of  the  battery  diminishes.  The  absolute  work  which  the 
machine  can  perform  in  a  given  time  is  therefore  diminished,  but  in  the  same 
time  the  heat  generated  by  the  decomposition  of  a  given  quantity  of  zinc 
diminishes  also  ;  the  fraction  of  the  work  of  the  chemical  forces  which  is 
transformed  into  heat  diminishes,  and  the  portion  equivalent  to  the  work  of 
the  machine  approaches  unity  the  more  the  velocity  increases.  We  may, 
therefore,  by  a  suitable  increase  of  the  velocity,  transform  with  any  desired 
completeness  the  entire  work  of  the  chemical  action,  or,  what  is  the  same 
thing,  the  entire  heat  generated,  into  mechanical  work.:]: 

Thus  the  electro-magnetic  engine,  which  has  so  far  proved  in  practice  the 
most  imperfect  of  all  engines,  is  theoretically  the  most  perfect  and  efficient. 
It  only  can  utilize  all  the  heat.     It  does  not,  however,  follow  that  it  is  only 

*  See  Note  20,  at  the  end  of  these  lectures.  t  See  Note  21.  %  Sec  Note  22. 


INTRODUCTION.  43 

to  conform  to  tlie  requirements  of  theory,  tliat  is,  to  give  the  engine 
the  greatest  possible  velocity,  in  order  to  make  it  practically  available.  The 
zinc  and  acids  necessary  for  the  battery  are  expensive,  more  so  than  that  of  the 
carbon  used  with  them,  and  in  spite  of  the  theoretical  superiority  of  the  electro- 
magnetic engine,  it  is  much  more  economical  to  consume  this  carbon  as  fuel  in 
a  steam  or  hot-air  engine.  This  must  be  the  case  until  we  can  obtain  with  less 
cost  bodies  which  possess  powerful  chemical  affinities,  that  is,  substances  which 
strive  energetically  to  remain  in  chemical  union,  or  to  return  into  union.  The 
solution  of  such  a  problem  appears  not  much  more  probable  than  the  discovery 
of  deposits  of  native  zinc,  or  of  springs  of  sulphuric  acid. 


VIII. 

We  have  by  no  means  exhausted  all  that  may  be  learned  from  the  electro- 
magnetic engine.  We  may  arrive  at  results  not  less  important  than  the  pre- 
ceding by  assuming  the  oi'dinary  action  of  the  engine  to  be  reversed,  so  that 
work  is  consumed  instead  of  being  produced. 

If,  for  example,  the  current  passes  through  the  fixed  spiral,  and  we  unite 
the  ends  of  the  moving  spirals  by  a  conducting  wire,  so  that  one  or  more  circuits 
are  formed,  the  machine  cannot  be  set  in  action  without  causing  induction 
currents  in  these  circuits. 

These  induced  currents  oppose  a  resistance  to  the  motion  of  the  machine, 
which  increases  the  amount  of  that  work  necessary  in  order  to  work  the  machine 
with  constant  velocity.  In  the  same  time,  the  wires  traversed  by  the  induction 
current  are  heated,  and  the  final  result  is  the  transformation  of  a  certain 
amount  of  work  into  heat.  The  determination  of  these  two  quantities  furnishes 
a  new  value  for  the  mechanical  equivalent,  J. 

It  was  with  such  experiments  that  Joule,  in  1843,  began  his  researches. 
He  deduced  a  value  for  J  of  452,  deviating  considerably,  therefore,  from  the 
later  and  more  accurate  determinations.  But,  hov/ever  considerable  the  differ- 
ence may  appear,  we  may  safely  assume  that  it  can  be  entirely  attributed  to 
the  difficulties  of  the  determination  and  the  imperfection  of  the  apparatus.* 

If  we  substitute  for  the  wire  spirals,  composed  of  wire  of  greater  or  less 
length  and  fineness,  in  which  the  induction  current  is  of  small  intensity,  and 
therefore  generates  but  little  heat,  a  metallic  plate  of  0.01  meter  in  thickness, 
and  of  a  diameter  corresponding  to  the  dimensions  of  the  fixed  electro-magnet, 
we  shall  find  that  the  current  is  greatly  increased  and  that  a  great  amount  of 
heat  is  generated.  There  must,  therefore,  be  considerable  expenditure  of 
work. 

This  new  form  of  the  experiment  is  interesting  on  two  accounts.  From  a 
theoretical  point  of  view  we  notice  that  the  principle  of  the  equivalence  of  work 
and  heat  gives  in  this  case  a  direct  relation  between  the  beginning  and  end  of  a 
series  of  effects,  the  intermediate  portions  of  which  are  at  present  but  little 
understood,  which  are  very  different  in  conductors  of  considerable  dimensions 
from  simple  wires.  In  the  second  place,  the  heat  generation  is  so  considerable 
that  it  is  possible  to  make  it  evident  by  instruments  of  suitable  delicacy,  and 
show  it  to  a  room  full  of  persons. 


44 


LECTURE  11. 


Some  of  you  bave,  witbout  doubt,  recognized  in  tbese  remarks  tbe  experi- 
ment of  Foucault,  wbicb  attracted  so  much  attention.  In  order  to  derive  from 
it  all  the  information  possible,  it  will  be  advantageous  to  give  it  two  different 
forms. 

First  we  give,  by  some  simple  mechanism,  the  metallic  plate  a  great  veloc- 
ity, without  allowing  the  current  to  pass  through  the  conducting  wire  of  the 
electro-magnet  between  whose  poles  the  plate  revolves.  As  soon  as  the  desired 
velocity  is  attained,  we  let  the  current  pass,  and  the  plate,  by  the  influence  of 
the  induced  currents,  is  brought  instantly  to  rest.  The  induced  currents  cease 
as  soon  as  the  plate  comes  to  rest,  but  the  heat  remains.  We  may  say  that  the 
final  result  of  the  experiment  is  that  all  the  living  forces  which  before  belonged 
to  the  entire  mass  have  been  transferred  to  the  molecules  and  become  visible 
as  heat.  The  sudden  stoppage  of  the  plate  can  be  easily  proved,  and  shows 
plainly  the  existence  and  intensity  of  the  induced  currents,  but  by  reason  of 
the  magnitude  of  the  number  which  expresses  the  mechanical  equivalent,  the 
heat  developed  is  but  slight  and  can  only  be  made  evident  by  the  most  sensitive 
instruments.  In  the  second  form  of  the  experiment  it  is  different.  We  let  the 
stream  pass  through  the  electro-magnet,  and  then  seek  to  put  the  plate  in  mo- 
tion. The  effort  necessary  for  this  is  a  visible  sign  of  the  resistance  to  be  over- 
come. A  few  minutes  of  motion  cause  a  rise  of  temperature  of  50  to  60  de- 
grees. 

IX. 

This  experiment  is  the  last  which  I  shall  borrow  from  physics.  Before 
leaving  the  domain  of  this  science  let  me  direct  your  attention  to  this  table,  in 
which  we  have  grouped  those  determinations  of  the  mechanical  equivalent 
which  are  the  most  reliable. 


TABLE   OF   THE   MECHANICAL  EQUIVALENT. 


Manner  of  Determination. 

Theoretical 

Principles 

pointed  out  by 

Experimental  Data 
given  by 

Value 

of 

Equivalent. 

General  properties  of  air 

Friction 

j  Mayer 
( Clausius 

Joule 

Clausius 

Joule 

Favre 

Bosscha 

Clausius 

5  Eegnault 

]  Molland  von  Beck 

Uoule 

\  Favre 

Hirn 

Joule 

Favre 
\  Weber 
1  Joule 

Quintus  Icilius 

U26 
435 

Work  of  steam  engine 

Heat  of  induced  currents 

Electro-magnetic  engines 

Daniell  battery 

413 

413 
452 
443 

U20 

400 

The  coincidence  of  these  results  is  very  satisfactory,  and  may  be  regarded  as 
confirmation  of  the  theory. 

There  are  only  two  values  which  deviate  greatly,  the  value  452,  found  by  Joule 
in  his  experiments  with  induced  currents,  and  the  value  400,  given  by  Quintus 
Icilius.     I  have  already  pointed  out  the  imperfections  of  Joule's  determination. 


INTRODUCTION.  45 

As  to  the  value  found  by  Icilius,  it  is  suflBcieut  to  remark  that  his  methods  re- 
quire the  combination  of  a  great  number  of  accurate  determinations  which  are 
independent  of  each  other.  It  is,  therefore,  small  matter  of  wonder,  if  his  re- 
sult deviates  much  from  the  number  425,  which  seems  to  be  the  mean  of  the 
other  determinations.* 

Let  us  now  consider  the  application  of  the  new  theory  in  chemistry.  In  the 
three  kinds  of  machines  which  we  have  considered,  we  see  that  the  motive 
power  is  due  to  a  consumption  of  heat.  But  from  whence  comes  this  heat,  if 
not  from  the  action  of  chemical  forces?  In  the  steam  and  hot-air  engines  we 
have  heat  generated  by  the  chemical  action  of  combustion,  and  while  this  heat 
calls  into  existence  a  series  of  physical  processes,  a  part  of  it  disappears  and 
reappears  in  the  shape  of  mechanical  work. 

In  the  electro-magnetic  engine  the  transformation  is  direct.  The  heating 
effects  of  a  certain  amount  of  chemical  action  are  diminished  by  the  action  of 
the  opposed  "induced"  currents,  by  an  amount  exactly  equal  to  the  mechani- 
cal work  performed.  This  difference,  however,  cannot  conceal  the  funda- 
mental identity  of  the  three  cases.  In  all  three  the  motive  force  is  either  a 
direct  or  indirect  transformation  of  chemical  affinities. 

These  mysterious  forces,  which  seem  to  elude  all  exact  determination, 
thus  come  under  the  dominion  of  general  mechanical  laws,  and  are  susceptible 
of  numerical  determinations.  We  cannot  determine  their  actual  intensities, 
that  is,  measure  the  accelerations  wMiich,  in  a  given  time,  they  impart  to  the 
atoms  upon  which  they  act ;  but  their  work  in  the  composition  or  decomposi- 
tion of  any  combination  can  now  be  determined  with  the  same  exactness  as 
the  work  of  falling  water. 

If,  for  example,  one  gram  of  hydrogen  and  eight  grams  of  oxygen,  of  a  cer- 
tain temperature,  are  brought  together  under  such  circumstances  as  to  cause 
their  union,  and  if  we  then  bring  tlie  nine  grams  of  steam  or  water  back  to  the 
original  temperature,  the  amount  of  heat  which  must  be  imparted  to  outer 
bodies,  multiplied  by  the  mechanical  equivalent,  is  the  exact  work  of  the 
chemical  action,  provided  that  the  union  is  not  accompanied  by  any  outer 
work,  that  no  living  force  is  imparted  to  other  bodies,  or  imparted  to  the 
bodies  which  take  part  in  the  chemical  action.  The  case  of  an  explosion 
which  is  accompanied  by  mechanical  action  -is  therefore  excluded.  You  will 
readily  recognize  that  this  limitation  is  unavoidable,  for,  as  we  have  seen  in 
the  electro-magnetic  engine,  a  constant  amount  of  chemical  action  can  gen- 
erate a  different  amount  of  heat,  according  as  we  have  a  simultaneous  develop- 
ment of  mechanical  work  or  not. 

I  scarcely  need  to  call  your  attention  to  the  importance  of  this  new  point  of 
view  in  therino-chemical  investigations.  It  forms  at  once  a  bond  of  union  be- 
tw^een  chemistry  and  general  mechanics.  Nor  is  this  one  of  those  superficial 
and  unfruitful  remarks  which  are  constantly  uttered  as  to  the  tmiversality  of 
mechanical  laws  or  the  dependence  of  every  phenomenon  upon  motion.  We 
can  give  examples  of  chemical  action  which  can  only  now  be  perfectly  ex- 
plained by  mechanical  considerations.  Such  examples  are  found  in  that  part 
of  chemistry  which  we  may  call  electro-chemistry,  and  which  is  properly  con- 
sidered as  equally  belonging  to  chemistry  and  physics. 

*  See  Note  24. 


46  LECTURE  II. 

You  know  tlaat  a  current  which  passes  through  a  compound  conductor 
always  decomposes  it.  You  know,  further,  that  every  chemical  action  which 
occurs  between  two  conducting  bodies  which  form  a  closed  circuit  causes  a 
current.  Hence  it  seems  evident  that  there  must  be  decomposition  when  we 
put  the  poles  of  a  battery  in  connection  with  two  strips  of  platinum  immersed 
in  a  compound  conducting  liquid.  This  conclusion  is,  however,  inexact.  The 
decomposition  of  water,  for  example,  is  impossible  hy  means  of  a  battery  com- 
posed of  zinc  and  platinum  or  copper,  with  water  acidulated  with  sulphuric 
acid.  Ordinarily,  ia  order  to  increase  the  conductive  power  of  a  fluid,  acid  is 
added  to  it,  but  even  this  fails  here.  There  is  no  decomposition,  and  no  appre- 
ciable current  flows  through  the  apparatus.*  These  facts  appeared  for  a  long 
time  incomprehensible,  but  it  is  easy  to  show  that  it  is  mechanically  impossible 
under  these  circumstances  to  decompose  the  water.  The  negative  work  of  the 
chemical  aflSnities  is,  by  such  decomposition,  greater  than  the  positive  work  of 
the  affinities  in  the  battery. 

We  know  that  the  decomposition  of  one  equivalent  of  zinc  in  every  dilute 
acid  corresponds  to  37,360  heat  units.  On  the  other  hand,  the  combustion  of 
one  equivalent  of  hydrogen  generates  34,460  heat  units.  It  is  clear  that,  in 
the  battery  with  acid  water,  the  negative  work  of  the  chemical  aflani ties  must 
be  exactly  equal  and  opposed  to  the  positive  work  of  the  same  affinities  which 
in  an  apparatus  serve  for  the  combustion  of  the  hydrogen.-'  If,  therefore, 
according  to  the  laws  of  electro-chemistry,  we  assume  that  each  equivalent  of 
zinc  decomposed  in  the  battery  causes  the  decomposition  of  one  equivalent  of 
v.'ater,  or  the  formation  of  two  equivalents  of  hydrogen,  and  if  we  take  into 
account,  also,  the  heat  generated  in  the  conductor  by  the  current,  we  shall  find 
in  a  system  at  rest  a  greater  negative  than  positive  work,  and  at  the  same 
time,  in  addition,  a  generation  of  heat,  that  is,  of  living  force.  This  is  a 
mechanical  contradiction,  the  existence  of  which  shows  why  decomposition 
cannot  occur.     For  this  explanation  we  are  indebted  to  Favre.f 

Undoubtedly  this  phenomenon  appears  essentially  different  from  ordinary 
chemical  processes.  There  is  in  the  system  a  regular  combination  of  substances 
acting  upon  each  other,  and  also  conductors  which  take  no  part  in  the  chemi- 
cal actions,  and  yet  whose  presence  is  absolutely  necessary.  All  this  does 
not  seem  to  resemble  very  closely 'the  reactions  which  go  on  in  the  test  tube  of 
the  chemist.  If,  however,  we  recall  the  fact,  v/hich  is  to-day  settled,  that  the 
action  of  acid  hydrates  upon  metals  is  always  a  pure  galvanic  process,  in  which 
the  metal,  its  impurities,  and  the  acid  form  a  galvanic  chain,  we  shall  probably 
be  inclined  to  regard  the  difference  as  only  accidental,  and  to  see  in  this  first 
application  of  mechanical  considerations  to  electro-chemical  processes  the  type 
of  a  series  of  applications  which  may  in  future  extend  over  the  whole  domain 
of  chemistry.:}: 

Just  as  electro-chemical  phenomena  find  their  explanation  in  the  considera- 
tion of  the  heat  effects  of  combinations,  so  the  theory  of  electric  currents 
allows  in  many  cases  of  other  determinations,  in  the  place  of  calorimetric 
measurements,  which  may  be  made  much  easier  by  means  of  the  galvanometer 
and  the  rheostat.  From  Ohm's  laws,  together  with  the  laws  of  electric  heat- 
ing, we  deduce  that  the  total  heat  generated  by  the  chemical  action  involved 

*  See  Note  25.  t  See  Note  26.  X  See  Note  27. 


INTBOD  UCTIOX.  47 

in  tlie  decomposition  of  one  equivalent  of  zinc  is  proportional  to  that  number 
which  we  call  the  electro-motive  force.  This  at  least  is  always  the  case  when 
there  is  no  disturbing  action  by  reason  of  a  gas  developed  upon  the  surface  of 
a  metal  in  the  battery  or  in  the  circuit. 

This  relation,  first  clearly  announced  by  Helmholtz  in  1847,  appears  to  have 
been  discovered  by  Joule  in  1841.  It  gives  to  the  determination  of  electro- 
motive forces  especial  interest,  and  has  led  Kegnault  to  interesting  conclusions 
as  to  the  constitutions  of  metallic  amalgams.* 

We  have  long  known  that  the  electro-motive  force  of  a  galvanic  battery  is 
considerably  increased  when  we  substitute  amalgamated  for  pure  zinc,  but 
thus  far  have  been  able  to  give  for  this  peculiar  result  only  still  more  peculiar 
explanations.  Eegnault  remarked  that  amalgamated  zinc  generated  more 
heat,  in  combining  with  oxygen  and  an  acid,  than  pure  zinc,  and  that,  conse- 
quently, when  the  mercury  is  separated  from  the  zinc,  heat  is  developed.  The 
opposite  process — the  formation  of  the  amalgam — must  therefore  produce  cold. 
The  electro-motive  force,  on  the  other  haisd,  is  diminished  when  we  substitute 
amalgamated  for  pure  cadmium.  The  amalgamation  of  cadmium  must  there- 
fore produce  heat.  Both  these  conclusions  are  perfectly  confirmed  by  experi- 
ment. These  phenomena  find  their  explanation  in  the  almost  perfect  identity 
in  chemical  properties  of  zinc  and  cadmium,  and  their  great  difference  in  latent 
heat.  The  two  metals  possess  probably  almost  the  same  affinities  for  quick- 
silver, and  their  union  with  this  substance  must  therefore  involve  almost  equal 
quantities  of  heat.  When  the  amalgam  is  decomposed,  we  have  then,  as  the 
caloric  result,  only  the  difference  between  the  heat  developed  bj  chemical 
action,  and  that  absorbed  daring  decomposition.  We  thus  see  how  the  zinc 
can  produce  cold  and  the  cadmium  heat,  since  the  first  metal  requires  for 
liquefaction  about  twice  as  much  heat  as  the  second.  These  considerations 
apply  to  all  metallic  amalgams,  and  are  in  accordance  with  experiment.  (See 
the  remarks  of  Regnault  in  the  Comptes  rendus,  1860,  vol.  li.,  p.  778.) 


But  not  only  machines  owe  their  moving  force  to  the  work  of  chemical 
affinities.  The  motive  power  of  man  and  animals  is  due  to  the  same  cause. 
Breathing  and  all  the  chemical  reactions  which  take  place  between  the  outer 
atmosphere  and  the  body  serve  not  only  to  preserve  a  constant  temperature 
and  to  remove  waste  materials.  Breathing  is  also  the  source  of  the  capability 
which  a  living  being  possesses  of  moving  exterior  bodies,  or,  by  means  of  some 
outer  point  of  resistance,  of  moving  itself. 

However  complex  these  chemical  reactions  may  be,  individually,  their  final 
result  corresponds  to  the  natural  tendency  of  the  affinities.  This  is  a  constant 
formation  of  water  and  carbonic  acid  at  the  expense  of  the  hydrogen  and 
carbon  which  exist,  either  in  the  body  or  in  the  food,  in  a  combination  in  which 
their  affinities  for  oxygen  are  not  completely  satisfied.  The  work  of  the 
chemical  processes  in  breathing  is  thus  essentially  positive.  If  the  animal  is 
at  rest,  this  work  is  the  equivalent  of  the  amount  of  heat  which  is  constantly 
developed  in  order  to  replace  the  loss  of  heat  due  to  radiation,  contact  with  the 


48  LECTURE  II. 

air,  and  perspiratiou.'"  If  the  animal  is  in  motion,  a  portion  of  the  work  of 
the  chemical  aflBnities  is  equivalent  to  the  outer  work  performed,  and  only  the 
remainder  is  converted  into  heat.  Therefore,  the  same  amount  of  chemical 
action  causes  in  an  organism  at  rest  a  greater  amount  of  heat  than  when  the 
organism  is  in  motion. 

These  ideas,  which  were  first  expressed  by  Julius  Robert  Mayer,  in  1845, 
have,  in  fact,  caused  in  general  physiology  an  advance  similar  to  tliat  which 
was  due  to  the  discoveries  of  Lavoisier  and  Senaebier  upon  respiration,  at  the 
end  of  the  last  century.  They  have  not  remained  purely  theoretical  ;  two 
different  series  of  experiments  have  given  very  remarkable  confirmation.  Tlie 
first  is  due  to  Hirn.  It  consisted  in  confining  a  man  in  a  closed  room,  who  for 
some  time  remained  at  rest.  He  then  performed,  for  some  time,  woi'k  in  a 
tread-mill,  and  in  both  cases  the  beat  and  chemical  action  of  respiration  were 
observed.  Each  time  the  heat  generated  and  the  carbonic  acid  exhaled  were 
measured.  The  ratio  of  the  first  to  the  second  was  less  during  motion  than 
during  rest.  A  given  amount  of  chemical  action,  therefore,  developed  less 
heat  while  work  was  performed  than  during  rest.  The  difference,  for  each 
individual,  was  closely  proportional  to  the  work  performed.  The  conditions  of 
the  experiment  are,  however,  too  complex,  and  the  changes  in  the  body  too 
difficult  of  measurement,  in  order  to  determine  in  this  way,  as  Hirn  endeavored 
to  do,  the  value  of  the  mechanical  equivalent  of  heat. 

Beclard  attacked  the  question  in  another  way,  and  by  means  of  an  experi- 
ment which  any  one  who  possesses  a  good  thermometer  can  repeat,  showed 
that  heat,  in  the  organism,  is  transformed  into  work.  By  the  simple  applica- 
cation  of  such  an  instrument  to  the  muscles  of  the  arm,  he  show^ed  tliat  the 
heat  generated  by  muscular  contraction  was  always  less  when  the  muscles, 
during  contraction,  performed  outer  work,  such  as  the  raising  of  a  weight. 
He  showed,  further,  that  the  heat,  on  the  other  hand,  is  increased  wben  the 
muscles  support  a  weight,  which,  falling  under  the  action  of  gravity,  performs 
outer  positive  work. 

The  results  of  these  two  series  of  experiments  are  the  most  valuable  by 
which  experimental  physiology  has  in  recent  times  been  enriched.  It  is,  more- 
over, clear  that  they  are  in  nowise  contradicted  by  the  daily  experience  that 
every  bodily  exertion  is  accompanied  by  heat.  The  contraction  of  a  muscle 
increases  undoubtedly  the  heat  generated  in  an  organism  in  a  given  time,  but  it 
increases  also  the  consumption  by  respiration,  so  that,  even  without  direct  proof, 
we  might  conclude  the  necessity  of  nourishment  as  a  consequence  of  work. 

The  investigations  of  Hirn  and  Beclard  simply  show  that,  in  accordance 
with  the  theory  of  Mayer,  the  consumption  increases  in  a  greater  ratio  than 
the  generation  of  heat.  Every  animal,  every  being  endowed  with  voluntary 
motion,  can  thus  be  regarded  as  a  heat  engine.  Every  motion  is  but  a  partial 
transformation  into  work  of  heat,  furnished  by  combustion  of  fuel  in  the  shape 
of  food,  etc.,  perfectly  comparable  to  that  transformation  which  occurs  in  the 
electro -magnetic  engine.  If  a  living  being  can  apparently  increase  at  will  the 
sum  of  the  living  forces  surrounding  him  at  any  moment,  it  is  only  under  the 
condition  that  the  sum  of  the  living  forces  of  the  heat  generated  by  chemical 
actions  in  his   own  organism  shall  be  diminished  by  a  precisely  equivalent 

*  Sec  Note  29. 


INTRODUCTION.  49 

amount.  We  may,  in  fact,  say  tliat  lie  pc>ssesses  only  tlie  power  of  directing 
the  living  forces  which  are  being  constantly  generated  witbiu  him  by  the 
action  of  clieniical  affinities  ;  and  in  order  to  show  the  true  nature  of  this 
power  I  cannot  do  better  than  to  borrow  from  Mayer  the  comparison  of  the 
will  in  a  sentient  animal  to  the  man  at  the  helm  of  a  steamship,  which  lie 
indeed  guides,  but  without  being  in  any  sense  the  physical  cause  of  the 
motion.  "The  motion  of  the  vessel,"  says  Mayer,  "obeys  the  will  of  the 
pilot  and  the  engine-driver — the  spiritual  influence  without  which  the  ship 
could  not  move,  or  would  be  destroyed  on  the  first  reef.  The  pilot  guides,  but 
he  does  not  move.  For  motion,  the  force  in  the  coal  is  necessary,  and  with- 
out this  the  ship  must  remain  at  rest — dead — no  matter  how  strong  the  will 
of  the  pilot." 

XL 

An  essentially  different  field  is  opened  to  us  by  the  vegetable  kingdom.  In 
the  higher  plants,  at  least,  the  final  result  of  the  life  processes  is  opposed  to 
the  chemical  affinities.  Under  conditions  which  continually  tend  to  convert  sub- 
stances into  carbonic  acid  and  water,  still  the  higher  plants  are  constantly  in- 
creasing the  quantity  of  these  substances  already  existing.  The  work  of  the 
affinities  within  them  is  therefore  a  negative  one,  and  our,  as  yet,  complete  ignor- 
ance as  to  the  mechanism  of  plant  life  need  not  jirevent  our  giving  complete 
assent  to  this  conclusion,  for  it  is,  after  all,  only  formulating  that  which  goes  on 
in  every  forest  and  on  every  meadow.  Apparently  without  sustenance,  and 
year  after  year,  wood  and  grasses  are  produced,  and  removed  by  man.  But  this 
continual  triumph  of  the  vegetable  kingdom  over  chemical  affinities  can  only 
be  sustained  by  an  equivalent  consumption  of  living  force  or  of  heat.  Hence, 
for  all  vegetation,  the  direct  or  indirect  action  of  the  sun  is  an  absolute  neces- 
sity. Only  infusorial  plants  and  parasites  are  exceptions.*  Neither  the  special 
refrangibility  of  those  rays  which  are  considered  as  especially  favorable  to 
vegetation, f  nor  the  weakness  of  their  thermometric  action,  distinguishes  these 
rays  essentially  from  those  which  are  called  "heat  rays."  That  which  the 
plant  absorbs  from  the  sun  is  heat — that  is,  living  force  ;  a  vibrating  motion 
of  atoms  distinguished  only  by  period  and  amplitude  of  vibration  from  those 
which  act  upon  the  thermometer.  By  the  consumption  of  this  living  force  the 
amount  of  combustible  material  is  increased.  In  the  combustion  of  the  pro- 
ducts of  vegetation  we  simply  recover  this  living  force,  which  opposed  the 
chemical  affinities  and  prevented  combination.  Thus,  to  a  transformation  of 
the  sun's  heat  we  owe  the  fuel  we  burn  and  the  vegetable  food  which  sustains 
the  energies  of  man  and  beast.  To  a  similar  transformation  we  owe  all  the 
mineral  fuel  which  sustains  our  industries.  When  we  remember,  further, 
that  it  is  the  sun  which  makes  the  wind  to  blow,  which  evaporates  water,  and 
causes  rain  and  sustains  rivers,  you  will  recognize  that  not  only  the  motion  of 
the  tides,  but  every  motion  upon  this  earth,  has  its  origin  directly  or  indirectly 
in  the  sun. 

*  See  Note  .30. 

t  See  Note  31.    Verdet  wrote  the  above  lines  under  the  erroneous  impression  that  the 
move  refrangible  part  of  the  spectrum  was  that  which  furnislied  the  livin^  force  to  the  plant. 
The  Note  31  gives  those  experiments  and  views  which  are  now  generally  accepted. 
4 


50  LECTURE  II. 


XII. 


This  beautiful  natural  harmony  turns  our  attention  to  the  centre  of  our 
system,  and  leads  us  to  Ihe  consideration  of  the  astronomical  applications  of 
the  new  theory. 

You  are  all  familiar  with  the  hypothesis  of  Buffon  as  to  the  origin  of  the 
sun's  heat.  According  to  him,  the  unceasing  fall  of  comets  upon  the  surface 
of  the  sun  furnishes  the  materials  of  combustion.  The  more  we  have  learned 
about  the  motions  of  comets,  and  the  more  we  have  departed  from  the  idea 
that  the  sun  is  a  furnace  similar  to  our  artificial  sources  of  heat,  the  more  has 
Buffon's  hypothesis  been  forgotten.  The  mechanical  theory  of  heat  has  revived 
it  again  and  rejuvenated  it.  Mayer  first  called  attention  to  the  fact  that  a  body 
arriving  at  the  sun  would  lose,  at  the  moment  of  its  impact,  the  enormous 
quantity  of  living  force  due  to  the  action  of  gravity,  and  that  this  loss  of  living 
force  must  cause  a  development  of  heat.  It  is  sufficient,  therefore,  to  account 
for  all  the  heat  which  the  sun  gives  out  into  space,  if  its  mass  is  continually 
growing  by  the  addition  of  comets,  aerolites,  and  other  cosmical  bodies. 

William  Thompson,  who  has  developed  and  followed  up  this  idea  of  Mayer 
with  equal  sagacity  and  boldness,  believes  that  these  bodies,  which  by  their 
fall  heat  the  sun,  probably  come  from  that  immense  cloud  which  surrounds  it, 
to  which  astronomers  give  the  name  of  "zodiacal  light."  Assuming  this,  he 
was  able  to  calculate  the  mass  which  must  yearly  fall  upon  the  sun,  in  order  to 
compensate  the  loss  of  heat  which  follows,  from  the  researches  of  Pouillet 
upon  the  thermometrical  effects  of  the  sun's  rays.  If  this  mass  has  the  mean 
density  of  the  sun,  it  would  form  a  layer  of  only  twenty  meters  in  thickness. 
This  thickness  is  considerably  less,  if  we  assume  with  Watterson  that  the  mass 
which  the  sun  attracts  streams  in  from  all  quarters  of  space.  In  either  case 
there  is  an  inconsiderable  increase  of  diameter,  which  would  elude  the  closest 
scrutiny  for  many  years.  Even  according  to  Thompson's  estimate,  it  would 
require  no  less  than  four  hundred  years  in  order  that  the  angle  subtended  at 
the  sun  should  be  increased  by  one-tenth  of  a  second. 

But  another  consequence  of  the  hypothesis  allows  of  an  easier  test  by 
experiment.  The  sun  turns  upon  its  axis  in  about  25  days.  Every  foreign 
substance  which  unites  with  it  diminishes  its  velocity  of  rotation.  Thompson 
has  calculated  the  thickness  of  the  layers  gradually  deposited  upon  the  sun's 
surface,  and  found  that  the  retardation  would  amount  to  one  hour  in  53  years. 
Unfortunately,  we  cannot  at  present  determine  the  time  of  rotation  within  an 
hour.  It  is  very  difficult  of  determination,  as  it  must  be  found  by  observation 
of  sun  spots,  which  have  also  a  proper  motion  of  their  own,  as  well  as  sharing 
the  motion  of  the  sun.  The  influence  of  this  proper  motion  can  only  be  elim- 
inated by  long  series  of  observations.  This  second  confirmation  therefore  is  at 
present  impossible,  and  would  appear  to  be  so  for  a  long  time  to  come.  But  it 
is  not,  like  the  first,  deferred  to  an  infinite  time. 

The  fundamental  idea  of  the  mechanical  theory  of  heat  has  been  combined 
with  the  hypothesis  of  Laplace  as  to  the  origin  of  the  solar  system,  and  we 
thus  obtain  another  explanation  of  the  heat  of  the  sun  and  planets.  The 
endeavor  has  even  been  made  to  deduce  in  this  manner  the  age  of  the  sun. 
I  do  not  require  that  you  shall  follow  me  in  speculations  of  this  kind,  which 
may  perhaps  appear  as  extremely  hypothetical,  or  for  which  the  test  of  expe- 


INTRODUCTION.  51 

rience  appears  too  distant ;  but  I  would  point  out  how  far  the  scope  of  the  new 
theory  extends.*  In  this  connection  it  has  been  said  that  science  is  on  the 
road  to  the  discovery  of  new  laws,  as  fundamental  and  as  accurate  as  those 
made  known  by  Newton  in  his  generation.  You  may  perhaps  he  inclined  to 
see  nothing  chimerical  in  this  view. 


XIII. 

I  cannot,  however,  allow  you  to  believe  that  these  laws  are  yet  discovered. 
Now  that  I  have  indicated  all  that  the  new  theory  teaches  us,  I  must  also  call 
your  attention  to  what  it  compels  us  to  neglect.  The  principle  of  the  equivalence 
of  heat  and  work  is  only  one  form  of  the  equation  of  living  forces.  A  special 
advantage  connected  with  the  application  of  this  equation  is,  that  it  allows  us 
to  express  relations  between  different  conditions  of  the  same  system  which  are 
independent  of  intermediate  conditions.  Its  disadvantage  is,  that  it  shows  us 
nothing  as  to  these  intermediate  conditions.  This  is,  strictly,  the  true  charac- 
ter of  the  new  theory.  It  teaches  us  the  "  why  "  and  the  "how  much,"  but 
it  does  not  answer  the  "how."  Thus  we  know  indeed  that  steam,  in  expand- 
ing, transforms  a  part  of  the  heat  which  it  contains  into  work  or  living  force  ; 
we  know  that  induced  currents  are  necessary  in  order  that  both  motive  power 
and  heat  may  be  furnished  ;  but  in  both  cases  the  process  itself,  the  play  of  the 
elementary  forces,  is  unknown.  It  is  much  to  have  determined  the  true  nature 
of  a  problem,  and  to  have  confined  within  fixed  limits  the  field  opened  out  by 
hypothesis.  The  application  of  the  mechanical  theory  of  heat  to  gases  has 
led  directly  to  the  discovery  of  a  theory  of  their  constitution,  which  at  least 
expresses  very  satisfactorily  the  known  facts.  We  may  justly  hope  that  this 
will  not  be  the  only  example,  and  that  the  new  theory,  after  showing  the 
necessary  connection  between  phenomena,  will  also  aid  us  to  penetrate  into 
the  inmost  secrets  of  nature.    . 


XIV. 

The  importance  which  we  must  now  attribute  to  the  new  theory  renders  it 
necessary  as  well  as  desirable  that  I  should  give  a  short  history  of  it,  in  which 
I  shall  endeavor  to  do  justice  to  the  principal  discoverers.  This  is  so  much  the 
more  necessary,  inasmuch  as  I  have  thus  far  given  only  its  ideas  in  logical 
order,  without  reference  to  the  historical  sequence  of  the  discoveries. 

We  can  distinguish  two  periods  in  this  science.  In  the  first,  which  reaches 
up  to  the  year  1842,  similar  ideas  of  the  mechanical  heat  theory  were  held  by 
different  authors,  but  the  facts  thus  explained  were  soon  regarded  from  differ- 
ent points  of  view,  and  attempts  were  made  to  refer  them  to  general  laws. 
The  real  principle,  however,  was  not  discovered,  and  all  these  attempts  re- 
mained unfruitful  and  without  essential  influence  upon  the  progress  of  science. 
The  work  of  this  period  bore,  nevertheless,  its  fruits  in  due  time,  and,  as  often 
happens  with  great  discoveries,  was  revived  again  about  the  year  1842,  when 
several  geniuses  expressed  the  new  ideas  with  sharpness  and  clearness.     Shortly 


52  LECTURE  II. 

af::er  the  beginning  of  that  period  of  rapid  progress  which  always  attends  the 
discovery  of  a  true  principle,  a  few  years  sufficed  to  erect  the  shapely  struct- 
ure, which  I  have  endeavored  hastily  to  show  you  over. 

The  first  name  among  the  list  of  those  whom  we  can  call  the  forerunners 
of  the  mechanical  theory  of  heat,  is  the  famous  one  of  Daniel  Bernoulli.  The 
HydrodynamidI  of  this  great  geometer  and  physicist,  which  for  more  than  a 
century  remained  neglected  and  forgotten,  contains  the  theory  of  the  constitu- 
tion of  gases  to  which  I  have  already  referred.  His  contemporaries  probably 
saw  in  it  only  the  effects  of  the  old  Cartesian  hypothesis,  and  not  until  j'ecent 
times  has  it  been  recognized  that  here  was  the  germ  of  a  new  science. 

In  the  year  1780,  somewhat  more  than  forty  years  after  the  publication  of  the 
Hydrodynamics,  Lavoisier  and  Laplace,  while  speaking,  in  their  treatise  upon 
heat,"  of  the  two  hypotheses  which  can  be  formed  of  this  physical  agent,  ex- 
pressed themselves  as  follows  :    ' '  Other  physicists  believe  that  heat  is  only 

the  result  of   imperceptible  vibrations  of   matter In  any  system 

heat  is  the  living  force  of  the  imperceptible  vibrations  of  the  molecules  of  a 
body  ;  it  is  the  sum  of  the  products  of  the  masses  of  each  molecule  into  the 

square  of  the  velocities We  do  not  assume  to  decide  between  the 

two  hypotheses.  Many  phenomena  seem  to  sustain  the  last  ;  as,  for  example, 
the  heat  generated  by  the  friction  of  two  bodies  ;  but  there  are  others  which 
seem  better  explained  by  the  first.  It  may  be  that  both  hold  good."  But 
after  this  plain  and  clear  definition  we  find  nowhere  in  the  treatise  any  attempt 
to  compare  the  living  forces  of  heat  with  ordinary  living  force,  snch  as  the 
rotation  or  motion  of  the  center  of  gravity  of  a  body.  They  never  compare 
heat  with  anything  else  than  itself,  and  it  contributes  accordingly  little  to  the 
value  of  their  remarks  whether  heat  is  regarded  as  an  indestrnctible  matter 
or  as  a  quantity  of  living  force. 

Indeed  they  go  further,  and  regard  as  proved  a  principle  which  is  in  direct 
contradiction  to  that  of  the  transformation  of  heat  into  work.  "All  changes  of 
heat,"  they  say,  "  whether  actual  or  only  apparent,  which  any  system  of  bodies 
undergoes  while  changing  its  condition,  are  repeated  in  reverse  order  when 
the  system  is  brought  back  to  its  original  condition."  If  they  had  added,  p7'0- 
mded  such  change  of  conclition  is  not  accompanied  iy  outer  leorlc,  the  mechanical 
theory  of  heat  would  have  been  founded.  But  without  such  proviso  this  asser- 
tion of  Lavoisier  and  Laplace  is  an  error,  which  is  daily  confuted  by  the  steam 
or  electro-magnetic  engine. 

It  is  impossible  to  say  how  the  views  of  Lavoisier  upon  this  question  would 
have  changed  had  he  lived  longer.  We  conclude  from  his  treatise  upon 
chemistry,  that  up  to  the  year  1789  he  had  not  completely  given  up  the  theory 
that  heat  consists  in  a  motion  of  molecules. 

It  is  indeed  true,  that,  probably  in  deference  to  the  general  opinion,  he 
spoke  of  gases  as  if  they  were  composed  of  a  union  of  certain  bases  with 
caloric.  But  he  continually  made  limitations  of  which  we  find  no  trace  in  the 
writings  of  his  scholars,  and  it  was  not  without  some  hesitation  that  he  placed 
light  and  heat  at  the  head  of  the  list  of  simple  bodies. 

As  to  Laplace,  his  views  underwent  rapid  change.  In  all  which  he  wrote 
during  his  connection  with  Lavoisier  he  appeared  as  the  ardent  advocate  of 

-  Memoire  sur  la  clialcur.    Meinoires  de  rAcad^mie  des  Sciences,  1780,  p.  357. 


INTROD  VCTION.  53 

the  materiality  of  heat.     His  weighty  authority  alone  procured  belief  for  the 
theory,  which  rested  upon  not  the  slightest  proof. 

Toward  the  end  of  the  last  century,  about  the  years  1798  and  1799,  two 
experiments  were  made  which  sufficed  to  show  the  untenability  of  the  theory 
espoused  by  the  author  of  the  "Mecanique  Celeste."  These  were  the  famous 
experiments  of  Rumford  and  Davy  upon  heat  generated  by  friction.  Rumford, 
at  the  foundry  at  Munich,  measured  as  exactly  as  he  was  able  the  heat  gener- 
ated in  boring  a  cannon,  and  in  order  to  leave  no  doubt  as  to  its  origin,  he 
determined  the  specific  heat  of  the  bronze  and  of  the  borings.  There  appeared 
no  perceptible  difference  between  the  tv.-o,  and  thus  the  only  reasonable  expla- 
nation which  could  be  offered  by  the  material  theory  of  heat  was  negatived 
decisively. 

It  had,  in  fact,  been  assumed  that  in  the  pulverized  bodies  the  specific  heat 
was  much  less  than  in  the  same  bodies  solid,  and  it  followed  indeed  from  this 
assumption  that  the  pulverization  of  a  body  by  the  friction  must  be  accom- 
panied by  heat.  But  it  was  forgotten  that,  in  such  case,  friction  itself  must 
create  heat  when  there  is  no  change  in  the  rubbing  surfaces.  The  experiment 
of  Rumford  showed,  moreover,  the  incorrectness  of  any  such  assumption. 

The  experiment  of  Davy,  about  a  year  later,  showed,  if  possible,  still  more 
conclusively  the  error  of  the  old  theory.  Two  pieces  of  ice,  rubbed  together, 
melted  very  rapidly,  and  formed  a  liquid  whose  specific  heat  was  more  than 
double  that  of  the  ice.  Davy  also  used  every  precaution  in  order  to  show  that 
the  generation  of  heat  was  not  accompanied  by  any  noticeable  absorption  in 
any  part  of  the  apparatus. 

Among  the  contemporaries  of  Rumford  and  Davy,  Young  appears  to  have 
been  the  only  one  to  fully  realize  the  scope  of  these  experiments.  In  his  lec- 
tures on  natural  philosophy,  published  in  1807,  he  placed  them  among  his 
immortal  discoveries  as  to  the  nature  of  light,  and  he  all  but  reached  the  true 
principle  of  the  mechanical  heat  theory.  He  was  the  first  to  cast  doubt  upon 
the  principle  of  Lavoisier  and  Laplace,  to  which  I  have  alluded.  "  Probably 
not  in  a  single  case,"  he  says,  in  his  lecture  upon  the  nature  of  heat,  "is  the 
heat  absorbed  exactly  equal  to  the  heat  given  up  in  the  reverse  process."  In 
this  simple  doubt  lies  concealed  the  essential  principle  of  the  mechanical 
theory  of  heat.  Young,  indeed,  admitted  the  probability  of  the  equivalence 
of  the  absorbed  and  generated  heat,  but  the  simple  expression  of  doubt  upon 
an  axiom  of  this  character  in  the  year  1807  is  noteworthy.* 

Unfortunately,  this  was  tlie  period  when  the  law  of  double  refraction  was 
looked  upon  as  an  argument  in  favor  of  the  emission  theory  ;  the  same 
period  in  which  the  elegant  treatises  of  Fresnel  remained  forgotten.  Even 
when,  in  the  year  1824,  the  original  genius  of  Sadi  Carnot,  awakened  by  the 
industrial  revolution  inaugurated  by  the  steam  engine,  sought  to  unfold  the 
general  laws  of  heat,  he  accepted  without  question  the  materiality,  and  there- 
fore the  indestructibility,  of  heat  as  his  starting-point. f  It  may  perhaps  sur- 
prise ycu  when  I  add,  that  in  spite  of  this  fundamental  error,  the  names  of 
Sadi  Carnot  and  of  his  learned  commentator,  Clapeyron,  occupy  distinguished 

*  Lectures  on  Natural  Philosophy,  vol.  i..  p.  6.51,  edition  of  1867. 

\  Memoire  sur  la  puissance  motrice  de  la  chaleur.  Journal  de  TEcole  polytechnique,  vol. 
xiv.,  p.  170,  1834.    PoggendorfE's  Annalen,  vol.  59,  p.  446. 


54  LECTURE  11. 

places  in  the  history  of  the  science.  To  Sadi  Carnot  we  owe  the  methods  of 
discussion  of  which  the  mechanical  theory  of  heat  makes  use.  In  his  writ- 
ings we  find  the  first  examples  of  the  cycle  process — of  that  series  of  changes 
by  which  a  body  passes  in  a  determinate  manner  from  one  condition  to  another, 
and  then  in  another  determinate  manner  returns  to  its  initial  condition. 

Clapeyron  Las  cleared  up  the  obscurity  of  Carnot's  treatise,  and  showed 
how  to  treat  analytically  and  represent  geometrically  this  new  and  fruitful 
method  of  treatment.  These  two  have  to  a  certain  extent  created  the  logic 
of  the  science.  As  true  principles  were  discovered,  they  only  needed  to  be  sub- 
jected to  the  forms  of  this  logic,  and  it  is  easily  conceivable  that  without  the 
work  of  Carnot  and  of  Clapeyron  the  advance  of  the  new  theory  would  have 
been  much  less  rapid. 

Finally,  we  close  this  first  portion  of  our  historical  sketch  by  recalling 
that  Seguin,  in  a  work  published  in  1839,*  of  more  political  than  physical 
interest,  has  given  views,  as  to  the  steam  engine,  closely  related  to  those  by 
which,  in  our  first  lecture,  we  have  sought  to  render  plain  to  you  the  trans- 
formation of  heat  into  work.f 

I  come  now  to  tho^e  labors  by  which,  since  the  year  1842,  the  science  has 
been  built  up.  These  labors  are  more  especially  those  of  three  men,  who 
without  connection  with  one  another,  without  even  knowing  each  other,  at  the 
same  time  and  in  almost  the  same  manner  arrived  at  the  same  results.  The 
priority  in  sequence  of  publication  belongs  undoubtedly  to  the  German  physi- 
cian, Julius  Robert  Mayer,  whose  name  we  have  already  had  frequent  occasion 
to  mention,  and  it  is  interesting  to  note  that  it  was  through  observation  of 
facts  occurring  in  his  medical  practice  tiiat  he  was  led  to  recognize  a  necessary 
equivalence  between  heat  and  work.  The  changes  in  the  color  of  arterial  and 
venous  blood  turned  his  attention  to  the  theory  of  respiration. |  He  recognized 
at  once  the  breath  as  the  origin  of  the  motive  power  of  animals.  The  com- 
parison of  animals  to  heat  engines  led  him  gradually  to  the  discovery  of  the 
important  principle  with  which  his  name  is  forever  associated.  This,  at  least, 
is  the  account  which  he  himself  has  given  us  in  his  writings  of  the  develop- 
ment of  his  ideas. 

We  find,  moreover,  in  these  writings,  the  first  determination  of  the  mechani- 
cal equivalent  of  heat,  deduced,  in  perfect  accordance  with  principle,  from  the 
properties  of  gas*,  but  incorrect  in  value  because  the  true  values  of  the  coeffi- 
cient of  expansion  and  the  specific  heat  of  air  were  then  very  imperfectly 
determined.  Mayer's  papers,  "Die  organische  Bewegung  in  ihrem  Zusam- 
menhange  mit  dem  Stoffvvechsel,"  and  "  Beitrage  zur  Dynamik  des  Himmels," 
which  last  appeared  in  1848,  contained  also  physiological  and  astronomical 
applications  which  show  that  he  clearly  appreciated  the  scope  of  his  dis- 
covery. 

About  the  time  of  the  first  publication  of  Mayer,  a  series  of  articles  by 
Colding,  an  engineer  at  Copenhagen,  were  presented  to  the  Royal  Academy  of 
Sciences  at  Copenhagen,  which  contained  ideas  upon  the  power  of  steam  and 
hot-air  engines  very  similar  to  those  of  Mayer,  as  well  as  an  experimental 
determination  of  the  mechanical  equivalent  of  heat   by  friction,  which  does 

*  Etudes  sur  riufluence  des  chemms  de  fer,  p.  180.    Paris,  1839. 
t  See  Note  33.  %  See  Note  34. 


INTROB  UGTION.  55 

not  seem  to  have  been  very  exact.  This  is  sufBcieut  ground  for  including  his 
name  among  those  of  the  founders  of  the  new  theory.  But  it  is  readily  con- 
ceived that  the  contributions  of  this  physicist,  written  in  a  language  but  little 
known,  and  first  printed  several  years  after  their  receipt,  had  but  little  influence 
upon  the  development  of  the  science. 

The  third  discoverer  of  whom  I  have  to  speak,  Joule,  is  perhaps  the  one 
who  has  contributed  the  most  toward  the  proof  of  the  new  principles  and  their 
final  reception.  His  first  work  appeared  in  1843,  and  is  undoubtedly  later  than 
the  first  works  of  Mayer  and  Colding.  It  contains  experiments  upon  the  heat 
generated  by  induced  currents,  and  appears  to  have  excited  at  first  but  little 
notice.  His  experiments  in  1845  upon  the  heat  effects  of  expansion  and  con- 
traction of  gases,  were  the  first  to  give  him  the  fame  of  introducing  new  ideas 
into  science.  His  experiments  upon  friction  gave  the  first  reliable  determina- 
tion of  the  mechanical  equivalent  of  heat.  His  views  upon  gases  gave  the 
first,  and,  until  now,  only  complete  explanation  of  a  phenomenon  whose  laws 
could  be  laid  down  by  theory  without  disclosing  its  mechanism. 

Immediately  after  these  three  names  we  must  place  that  of  Helmholtz,  who, 
in  1847,  in  his  article,  "  The  Conservation  of  Force,"  first  united  the  new  ideas 
into  a  complete  structure,  and  drew  from  them  fruitful  and  important  applica- 
tions to  induction  phenomena,  electro-chemistry,  thermo-electric  currents,  etc. 

The  development  proper  of  the  mechanical  theory  of  heat  as  a  science,  the 
clear  and  methodical  presentation  of  methods  of  investigation  and  discussion, 
and,  finally,  their  application  to  machines,  is  due  to  four  savaus,  whose  names 
are  the  last  that  I  shall  mention  ;  Clausius,  Macquorn  Rankine,  William 
Thompson,  and  Gustav  Zeuner.  Their  most  important  investigations  extend 
from  1849  to  the  present  day. 

Many  other  workers  might  be  mentioned.  I  have  already  had  occasion  to 
notice  several  in  the  course  of  these  lectures  ;  I  will  not  seek  to  extend  the  list, 
but  shall  content  myself  with  the  names  of  those  investigators  who  have  laid 
the  foundation  stones  of  the  edifice,  upon  whose  completion  so  many  have  for 
the  last  thirty  years  labored  with  such  signal  success. 


NOTES 


ADDITIONS    TO    THE    LECTUKES. 


CONTENTS. 


1.  Upon  perpetual  motion. — 2.  Upon  the  origin  of  the  motive  power  in  the  steam  engine 
under  the  hypothesis  of  the  materiality  of  heat.— 3.  Some  experiments  of  Him  which  apparently 
contradict  theory. — 4.  Upon  a  theorem  of  Coriolis.— 5.  Upon  the  law  of  expansion  of  gases.-- 
6.  Upon  the  inner  work  in  crystals  and  some  liquids.— T.  Upon  an  incorrect  determination  of  the 
mechanical  equivalent  of  heat.— 8.  Upon  bodies  which  contract  under  the  action  of  heat.— 
9.  Calorimetric  measurements  in  which  the  outer  work  has  not  been  considered.— 10.  Theory  of 
the  constitution  of  gases. — 11.  How  gases  and  vapors  perform  outer  work. — 12.  Upon  the  value 
of  the  mechanical  equivalent  given  by  carbolic  acid.— 13.  Principle  of  the  experiments  of  Thom- 
son and  Joule  upon  the  heat  phenomena  of  moving  gas.— 14.  Upon  the  condensation  which 
accompanies  the  expansion  of  steam. — 1.5.  Upon  the  regenerator  in  gas  engines. — 10.  Determina- 
tion of  the  economical  coefficient  for  Ericsson's  engine  and  for  the  engine  without  regenerator. 
■ — 17.  Upon  gas  engines  in  which  the  temperature  sinks  to  the  absolute  zero. — 18.  Upon  the 
necessity  of  the  tendency  of  heat  to  pass  from  a  warmer  to  a  colder  body.— 19.  Upon  the  role 
played  by  friction  in  the  electro-chemical  investigations  of  Favre.— 20.  The  discovery  of  induc- 
tion phenomena.— 21.  The  deduction  of  induction  phenomena  from  theory.— 22.  Upon  the  com- 
plete transformation  of  heat  into  work  by  the  electro-magnetic  engine.— 24.  Determination  of 
the  mechanical  equivalent  of  heat  by  the  electro-magnetic  engine  (Joule).— 24.  Upon  the  nature 
of  the  electro-magnetic  and  electro-dynamic  forces.— 23.  Electrolytic  convection.— 26.  Upon  the 
polarization  of  the  electrodes.— 27.  The  decomposition  of  zinc  in  dilute  acids.— 28.  Upon  the 
application  of  the  measurement  of  electro-motive  forces  to  thermo-chemical  investigations. — 
29.  The  influence  of  the  friction  of  the  blood  upon  animal  heat.— 30.  Upon  vegetation  which 
takes  place  in  the  absence  of  light.— 31.  The  absorption  spectrum  of  chlorophyl  and  the  influence 
of  the  color  of  light  upon  the  growth  of  plants.— 32.  Remarks  of  Mayer  upon  the  ebb  and  flow 
of  the  tides.— 33.  Upon  a  demonstration  of  Seguin  relating  to  the  steam  engine.— 34.  The  depend- 
ence of  the  color  of  venous  blood  upon  the  temperature. 

Addition.— The  entropy  of  the  world  tends  towards  a  maximum. 


NOTE  1.— (Pages.) 

THE   PEOBLEM    OF   PERPETUAL   MOTION". 

In  accordance  with  usual  custom,  I  have  shown  the  impossibility  of  per- 
petual motion  to  be  a  consequence  of  the  fundamental  principles  of  mechanics, 
as  well  as  of  the  manner  in  which  the  forces  of  nature  act. 

We  may,  however,  also  recognize  in  this  an  independent  and  apparently  self- 
existent  principle,  which,  at  bottom,  expresses  nothing  else  than  the  necessity 
of  a  definite  relation  between  cause  and  effect. 

Considered  thus,  the  principle  of  the  impossibility  of  perpetaal  motion  may 
be  used  to  prove  that  all  natural  forces  must  act  in  the  line  joining  any  two 
mutually  interacting  material  points,  and  that  these  mutual  actions  are  func- 
tions of  the  distances  apart. 

This  is  the  method  adopted  by  Helmholtz  in  his  famous  Treatise  "  Die 
Erhaltung  Kraft  "  (Berlin,  1847) — a  method  which  may  seem  to  many  the  best. 
Helmholtz  says  : 

"  Let  us  consider,  first,  a  material  point  of  the  mass  m,  which  moves  under 

the  influence  of  the  forces  of  any  number  of  bodies  composing  a  fixed  system 

A,  then  we  can  determine  for  every  moment  the  position  and  velocity  of  this 

point.     Let  the  time  t  be  the  primitive  variable,  and,  dependent  upon  it,  let  the 

ordinates  of  m,  with  reference  to  a  co-ordinate  system  fixed  with  respect  to  the 

system  A,  be  «,  y,  and  s,  the  tangential  velocity  be  q,  the  three  components  of 

the  same 

dx  dv  dz 

^'  =  ^'  ^  =  -17'  ^^=  57. 

dt  dt  dt 

and  finally  the  components  of  the  acting  forces 

^         dx  ^         dy  ^         dz 

X=m-r,  Y=m-£,  Z=m-. 

dt  dt  dt 

Our  principle  requires  that  ^mg-,  and  hence  q" ,  shall  be  always  the  same 
when  m  has  the  same  position  with  respect  to  A,  and  therefore  not  only  a 
function  of  t,  but  also  a  function  of  the  co-ordinates  x,  y,  z,  alone,  that  is. 

Since  q-  =  u'^  +  v-  +  w- 

d  (g^)  =  2udu  +  2vd'o  +  2wdio. 

If  here,  instead  of  u,  we  put  — ,  instead  of  du,  —  ,  and  also  for  «  and  w 
^      dt  m 

similar  values,  we  have 

d{q'^)^^^dx  +  —dy  +  ^-^dz (2) 

m  m  m  ^ 

Since  equations  (1)  and  (2)  must  be  simultaneous  for  every  dx,  dy,  dz,  we 
have 

Htl  =  ^^  W)  _  2F  S (g-)  _  2Z 

8x  m  '  dy         m  '  dz  m 

If,  however,  g'  is  a  function  of  x,  y,  z,  it  follows  that  X,  Y,  Z,  that  is,  the 

59 


60  NOTES  AND  ADDITIONS 

direction  and  intensity  of  the  acting  force  must  be  also  only  functions  of  the 
position  of  m  with  respect  to  A. 

If  we  conceive  instead  of  the  system  A,  a  single  material  point  a,  it  follows 
from  the  above  that  the  direction  and  intensity  of  the  force  which  a  exerts 
upon  m  is  determined  only  by  the  relative  position  of  m  with  respect  to  a. 
Since  now  the  position  of  m  is  determined  by  the  distance  between  m  and  a, 
in  this  case  the  direction  and  intensity  of  the  force  functions  must  be  this  dis- 
tance a.  If  we  conceive  the  co-ordinates  referred  to  any  system  whose  origin 
is  at  a,  we  must  have 

md{cif  =  2Xdx  +  2Ydy  +  2Zdz^0 (3) 

d  (r'^)  —  2xdx  +  2ydy  +  2zdz  —  0 

,            xdx  +  ydy 
dz  ^ . 

z 

This  value,  in  (3)  gives 

lx--z\dx+lY-^z\dy  =  0, 

hence  X=^Z     and'    Y=^Z, 

z  z 

that  is,  the  resultants  must  be  directed  towards  the  origin. 

Hence  in  a  system  which  is  subject  generally  to  the  law  of  the  conservation 
of  living  force,  the  simple  forces  of  the  material  points  must  be  central  forces. 


NOTE   2.— (Page  14.) 

THE    ORIGIN   OF   THE    MOTIVE    POWER   IN"   THE    STEAM   ENGINE   UPON 
THE   HYPOTHESIS   OF   THE   MATERIALITY    OF   HEAT. 

Sadi  Carnot,  assuming  the  materiality  of  heat,  has  given  an  explanation  of 
the  phenomena  in  the  steam  engine,  which,  although  it  does  not  agree  with 
reality,  is  not  so  evidently  erroneous  as  the  hypotheses  which  have  been 
framed  in  order,  by  the  same  hypothesis,  to  account  for  the  heat  generated  by 
friction. 

According  to  him,  the  imponderable  fluid  whose  presence  in  bodies  gives 
rise  to  those  various  effects  which  are  called  "  heat,"  has  an  inherent  tendency 
to  pass  from  a  hot  body  to  a  colder  one,  just  as  heavy  bodies  tend  to  fall  from 
a  high  place  to  a  lower ;  or  rather,  there  is  a  similar  tendency  due  to  the  action 
of  the  heat  molecules  on  each  other  and  the  actions  upon  them  of  the  ponder- 
able molecules.  Thus,  the  forces  which  act  upon  the  heat  molecules,  furnish 
positive  work  whenever  there  is  a  transfer  of  heat  from  a  hot  body  to  a  cold 
one,  which  cannot  indeed  be  a  priori  determined,  but  which  is,  however,  com- 
parable to  the  work  of  gravity  in  a  waterfall. 

This  is  the  true  motive  work  in  the  engine.  The  heat  from  the  boiler  to 
the  condenser  expei-iences  a  kind  of  fall  (this  is  Carnot's  expression),  and  the 
work  furnished  by  the  engine  is  the  equivalent  of  this  mechanical  process, 
just  as  the  work  of  a  water-wheel  is  the  equivalent  of  the  fall  of  the  stream. 

These  views  have  in  them  nothing  at  variance  with  common  sense,  or  which 
contradicts  the  general  view  of  the  phenomena  ;  but  it  is  evident  that  the  as- 
sumption of  the  materiality  of  heat  involves  that  of  its  indestructibility,  and 
hence,  in  the  case  of  the  steam  engine,  gives  rise  to  the  following  dilemma, 
the  solution  of  which  must  be  demanded  from  experience — either  heat  is  some- 
thing material,  and  then  the  steam  must  transfer  as  much  heat  to  the  con- 
denser as  it  takes  from  the  boiler — or  heat  is  a  motion  of  some  kind,  and  then 
a  part  of  the  heat  during  the  action  of  the  engine  must  disappear,  and  thus 
give  rise  to  outer  work. 

We  have  seen  what  answer  experience  has  given. 


TO   THE  LECTURES.  61 


NOTE  3.— (Page  14.) 

SOME     EXPERIMENTS    BY    HIEZsT    WHICH    APPiiRENTLY    CONTRADICT 

THEORY. 

The  investigations  of  Hirn  were  undertalien  in  consequence  of  a  prize  offered 
by  the  Physical  Society  of  Berlin  for  a  numerical  determination  of  the  true 
value  of  the  mechanical  equivalent  of  heat.  In  the  report  of  Clausius  to  the 
society,  he  calls  attention  to  the  error  of  Hirn's  views  upon  the  steam  engine, 
and  gives  a  correct  explanation  of  the  experiments. 

Hirn  did  not  agree  with  the  views  of  Clausius,  and,  although  when  his 
treatise  was  published,  he  gave  the  report  of  the  learned  physicist  in  full,  he 
sought  to  defend  the  correctness  of  his  first  calculations,  and  endeavored  to 
justify  them  by  two  difEerent  methods  of  experiment— viz.,  by  measuring  the 
heat  used  by  a  steam  engine  without  expansion,  and  by  investigating  the  heat 
phenomena  which  accompany  the  efflux  of  a  gas  under  high  pressure  into  a 
vacuum. 

It  may  not,  perhaps,  be  without  profit  to  show  wliat  the  value  of  these  new 
arguments  amounts  to. 

In  the  following,  we  give  the  words  in  which  Clausius  refers  to  the  incor- 
rectness of  Hirn's  views  : 

"It  can  be  easily  shown  how  this  error  of  Hirn  arises.*  He  says,  in  justi- 
fication of  his  assumption,  '  when  steam  condenses  under  the  same  pressure  at 
which  it  was  generated,  it  gives  up  during  condensation  as  much  heat  as  must 
have  been  imparted  in  its  generation.'  This  principle  is  indeed  correct,  but  it 
has  no  application  to  the  steam  engine. 

"  When,  in  an  engine  working  without  expansion,  the  steam  has  entirely 
filled  the  cylinder  back  of  the  piston,  and  then  the  con;munication  with  the 
condenser  is  opened,  only  at  first  does  the  steam  flow  under  full  pressure  into 
the  condenser,  and  then  the  in-essure  gradually  decreases  as  the  steam  stiil  in 
the  cylinder  expands.  By  this  expansion  the  steam  still  in  the  c.ylinder  is  con- 
siderably coded,  and,  if  not  superheated  or  heated  from  without,  will  be  partly 
■  condensed,  even  while  still  in  the  cylinder.  In  order  to  comply  with  the  con- 
ditions implied  in  the  above  principle,  the  piston,  during  the  efflux,  should  re- 
turn with  just  such  velocity  as  to  keep  the  steam  still  in  the  cylinder  always 
at  full  pressure.  But  then  the  back  pressure  would  have  to  be  as  great  as  the 
driving  pressure  was,  and  no  work  could  be  obtained.  If  the  author  had 
extended  his  experiments  to  engines  without  expansion,  he  would  undoubt-  . 
edly  have  found  for  these  also  that  the  amount  of  heat  given  up  is  less  than 
that  received." 

These  last  words,  without  doubt,  led  Hirn  to  make  an  experimental  in- 
vestigation of  a  steam  engine  without  expansion. f 

He  does  not  appear,  however,  in  this  new  investigation  to  have  succeeded 
in  overcoming  all  its  difficulties,  as  he  himself  says,  "  the  physicist  may  meet 
in  experimental  science  with  insurmotmtable  obstacles." 

He  says,  indeed,  that  he  has  established  that,  in  an  engine  without  expan- 
sion, the  heat  expenditure  is  either  zero  or  can  be  neglected,  but  together  with 
the  experiments  which  give  this  strange  result,  he  gives  the  data  of  another 
from  which  even  still  stranger  conclusions  may  be  drawn. 

In  an  engine  in  which  expansion  occurred  only  through  the  fifth  part  of  the 
stroke,  not  only  was  work  performed,  but  also  heat  generated. 

That  the  new  methods  of  experiment  which  lead  to  such  conclusions  are 
to  be  preferred  to  those  used  by  Hirn  in  his  first  experiments,  is  more  than 
doubtful. 

*  Fortschritte  der  Physik,  1855.  Bd.  xi.,  p.  21. 

+  Recherches  sur  reqaivalent  mecauique  de  la  clialeiir,  par  Gustave  Adnlplic  Hiin,  Paris, 
1858,  p.  179. 


62  NOTES  AND  ADDITIONS 

The  clear  and  decisive  criticism  of  Clausius  holds  in  full  force. 

Hirn  opposes  to  Clausius  also  the  follovviug  expeiiuienr  :  Into  a  receiver  of 
black  lead,  surrounded  by  cold  water,  he  allowed  a  jet  of  steam  under  high 
pressure  to  enter,  the  temperature  of  which  was  measured  by  a  tbermometer 
just  before  the  steam  reached  the  orifice. 

He  then  collected  the  water  condensed  in  a  given  time,  and  from  the  rise  of 
temperature  of  the  calorimeter,  applying  the  necessary  corrections,  found  the 
heat  given  up  during  condensation. 

In  this  way  a  greater  number  was  always  found  than  that  given  by  the  ex- 
pression 

p  [60().5  +  '0.305f  ^  0.4805  {T -  i)  -  z], 

which  gives  the  heat  contained  in  the  steam  at  the  moment  at  which  it  arrives 
at  the  orifice— where  p  is  the  weight  of  the  steam,  Zits  actual  temperature,  t 
the  temperature  at  which,  under  the  actual  pressure,  it  would  be  saturated, 
and  r  the  temperature  of  the  condensed  water,  assuming,  according  to  Reg- 
nault's  experiments,  that  0-4805  is  the  specific  heat  of  the  steam. 

Similar  experiments,  in  which  he  used  the  condenser  of  a  steam  engine  as  a 
calorimeter,  gave  the  same  result. 

Hirn  concluded  that  saturated  or  superheated  steam  which  condenses  in  a 
cooling  vessel  in  wiiich  the  pressure  is  less  than  the  initial  pressure  of  the 
steam  generates  heat. 

The  fact  is  remarkable  and  interesting,  but  easily  to  be  accounted  for. 

The  steam  which  leaves  the  orifice  of  efflux  has  a  very  great  velocity,  the 
liquid  which  results  from  the  condensation  is  at  rest.  In  the  transformation 
from  the  gaseous  to  the  liquid  condition,  then,  a  large  amount  of  living  force 
disappears,  and  there  is,  therefore,  according  to  the  new  principles,  a  genera- 
tion of  heat. 

It  is,  indeed,  true  that  the  outer  work  performed  on  the  steam  during  its 
condensation  is  less  t^an  that  wliich  it  performed  during  its  generation,  and 
this  diminishes  the  heat  generated  during  condensation,  but  there  is  no  exact 
compensation.  If,  therefore,  the  steam  which  enters  the  calorimetric  appara- 
tus is  saturated  steam  of  five  atmospheres,  we  must  impart  to  each  unit  of 
weight  of  water  of  the  temperature  r,  in  order  to  generate  it,  651  — r  heat  units. 

A  part,  q  ,  of  this  heat  goes  to  increase  the  living  force  of  the  molecules.  A 
second  part,  q' ,  corresponds  to  the  change  of  aggregation  or  disgregation  work.  ■ 
A  third  part,  q" ,  is  the  equivalent  of  the  outer  work. 

The  last  part,  </'",  can  be  taken  equal  to  nearly  44  heat  units,  if  we  take  for 
the  absolute  density  of  saturated  steam  of  one  atmosphere  the  value  ji^t,  theo- 
retically determined  by  Clausius,*  and  if  we  neglect  the  very  small  difference 
between  the  volume  of  water  at  r  degrees  and  at  zero. 

On  the  other  hand,  the  recent  labors  of  Minary  and  Resalf  enable  us  to  de- 
termine the  weight  of  steam  which  will  flow  in  five  minutes  from  a  boiler  under 
five  atmospheres'  pressure,  through  .an  orifice  of  0.007  meter  diameter,  when 
this  orifice  is  at  the  end  of  a  pipe  of  0.15  meter  diameter.  This  steam  weight 
is  10.0  kilograms. 

Hence  we  can  easily  find,  with  the  preceding  value  for  the  density,  and 
taking  for  the  coefficient  of  contraction  0,44  (a  value  given  by  the  experimenter) 
that  the  velocity  of  efflux  is  about  600  meters  per  second,  and  hence  that  every 
kilogram  of  steam  which  issued  in  Hirn's  experiment  carried  with  it  a  living 
force  of  about  180,000  meter-kilograms,  equivalent  to  about  400  heat  units. 
We  see,  therefore,  that  even  without  outer  work  there  is  more  than  compensa- 
tion, and  that  the  disappearance  of  the  living-  force  is  more  than  sufficient  to 
explain  the  phenomena  observed  by  Hirn.  Even  a  considerable  error  in  the 
coeflacient  of  contraction  would  not  affect  the  conclusion. 

It  is  worth  remarking  that  the  living  force  which  the  steam  possesses  when 
it  leaves  the  orifice,  is  itself  a  transformation  of  the  heat  which  the  boiler  pos- 

*  The  theoretical  vahies  of  Clausiui?  (Alihandhmgeii,  Bd.  i.,  p.  72)  .-ire  confirmed  by  the  ex- 
periments of  Pairbairn  and  Tate.  (Proc.  of  the  Royal  Soc,  ISbO,  in  Phil.  Mag.,  4  iser.  vol.  xxi., 
p.  230,  and  Compter  Rendns,  vol.  Hi.,  p.  70li.) 

t  Annales  dcs  mines,  vol.  xviii.,  p.  ()53, 


TO   THE  LECTURES.  63 

sessed,  and  that  hence  the  steam,  at  the  moment  it  leaves  the  orifice,  can  no 
longer  be  in  the  same  condition  it  had  when  in  the  boiler  at  a  distance  from 
the  orifice. 


NOTE  4.— (Page  16.) 

UPOIT   A   THEOREM    BY    CORIOLIS. 

The  following  theorem,  given  by  Coriolis  in  his  classic  work  upon  the  cal- 
culation of  the  delivery  of  engines,*  is,  in  a  certain  sense,  an  illustration  of  the 
general  law  which  we  have  sought  to  enunciate  : 

"The  sum  of  the  living  force*  of  a  system  of  molecules,  wlmtever  may  be  the  kind  of  motion, 
may  be  divided  into  three  parts  : 

"  1.  The  living  force  of  all  the  molecules  when  concentrated  at  the  center  of  gravity  of  the 
system. 

'■  '2.  The  sum  of  the  living  forces  of  these  molecules,  when  we  assume  that  they  constitute,  in 
the  same  relative  positions  in  which  they  occur,  a  body  of  invariable  form  to  whicli  is  imparted 
the  mean  motion  nliout  the  center  of  gra\'ity. 

'■3.  Tlie  sum  of  the  living  forces  of  the  molecules,  by  reason  of  the  relative  velocities  which 
they  possess  with  reference  to  co-ordinate  planes  which  partake  of  the  mean  motion  of  rotation."' 

In  the  equation  of  works  we  have  usually  to  take  account  only  of  the  two 
first  portions,  that  is.  the  living  forces  due  to  the  rectilinear  and  rotary  motion 
of  the  bocjy.  The  third  portion  is  usually  submitted  to  calculation  only  when 
the  vibrations  are  sensible  or  apparent,  as  in  the  case  of  sound  vibrations. 

The  fundamental  idea  of  the  new  theory  is  to  seek  this  third  part  in  the 
heat. 

It  is,  moreover,  evident  that  the  action  of  mechanical  forces  in  most  cases 
Avill  give  rise  to  all  three  kinds  of  living  forces,  and  that  we  have  just  as  little 
reason  to  neglect  the  changes  of  the  living  force  of  heat  as  of  the  outwardly 
visible  living  forces.  We  may  even  add  that  the  transformation  of  the  out- 
wardly visible  living  forces  into  the  living  force  of  heat  takes  place  incessantly 
in  nature  before  our  eyes,  and  that  it  is  chiefiy  in  this  way  that  the  vibrations 
of  a  system  about  a  stable  position  of  equilibriutn  are  extinguished. 


NOTE  5. -(Page  17.) 

IHE   LAW   OF   EXPANSIO]Sr    OF   GASES. 

The  law  of  expansion  of  gases  was  held  by  all  physicists,  down  to  Magnus 
and  Eegnault,  as  exact.     It  is  generally  known  as  the  law  of  Gay-Lussac. 

It  is,'  in  my  opinion,  more  correct  to  call  it  the  law  of  Charles.  The  essen- 
tial part  of  this  law,  viz,,  the  approximate  agreement  of  the  expansion  of 
different  gases,  and  hence  the  proportionality  of  all  these  expansions  to  the 
temperature,  as  determined  by  a  thermometer,  which  is  itself  formed  by  a  gas, 
was  pi-oved  by  Charles  in  the  simplest  manner. 

The  reservoir  of  a  kind  of  mercury  barometer  was  filled  with  gas.  The 
apparatus  was  exposed  successively  to  the  action  of  two  different  temperatures 
(the  ordinary  temperature  of  exterior  objects  and  that  of  boiling  water),  and 
the  rise  of  the  mercury  in  the  barometer  tube  observed.  Charles  found  that 
this  rise  for  air,  oxygen,  nitrogen,  hydrogen,  and  carbonic  acid  gas  was  the 
same,  and  no  more  was  needed  to  establish  the  fact  that  the  coefficient  of 
expansion  of  these  different  gases  is  nearly  the  same,  even  if,  in  this  manner, 
the  exact  value  of  the  common  coefficient  could  not  be  determined. f 

*  Coriolis,  Traite  de  la  mecauique  des  corps  solides  et  du  calcul  de  I'effet  des  machines,  2d 

t  The  experiments  of  Charles  are  mentioned  by  Gay-Lussac  himself  in  his  article  upon  the 
expan.sion  of  gases,  Ann,  de  Chim.,  vol.  xliii.,  p.  157. 


64  NOTES  AND  ADDITIONS 

Qay-Lussac  has  added  to  tliis  simply  a  determination  of  the  coeflBcient  of 
expansion,  which  was  inexact  by  about  nith. 

We  may  even  say  that  the  advance  or'  science  was  in  some  degree  retarded, 
in  that  he  regarded  as  ^n  absolute  law  that  which  was  only  an  approximate 
expression. 

According  to  Charles,  the  compound  gases  do  not  expand  as  much  as  those 
just  named.  It  is  not  known  exactly  to  what  gases  Charles  thus  refers,  but  it 
is  probable  to  the  same  as  those  experimented  upon  by  Gay-Lussac,  viz  , 
sulphurous  acid  and  hydrochloric  acid  gas,  for  which  he  gave  the  same  coetB- 
cient  as  for  air. 

We  know  now  that  the  coefficient  of  expansion  for  sulphurous  acid  is  jVth 
greater  than  for  air.  In  this  important  point,  then,  Charles  was  more  accurate 
than  Gaj -Luss.ac,  and  however  imperfect  his  method  of  experimenting  may  be 
considered,  it  Is  not  open  to  the  charge  of  not  being  able  to  distinguish  dif- 
ferences in  the  quantity  to  be  measured  of  iVth. 


NOTE  6.— (Page  18.) 

THE    DISGEEGATIOIT   WORK   IN"    CRYSTALS   AND    SOME   LIQUIDS. 

In  liquids  and  non -crystalline  solids  it  is  possible  that  for  a  simple  rise  of 
temperature  only,  when  there  is  no  change  of  volume,  there  is  no  disgregation 
work, 

It  is,  without  doubt,  essentially  different  for  crystalline  solids,  at  least  for 
those  belonging  to  the  tesseral  system  The  unequal  expansion  in  different 
directions,  caused  in  these  bodies  by  the  action  of  heat,  does  not  allow  the 
assumption  that,  when  the  expansion  is  prevented  by  a  sufficient  increase  of 
pressure,  there  is  no  change  in  the  arrangement  of  the  molecules. 

If,  for  example,  a  crystal  of  kalkspar  is  heated,  and  at  the  same  time  com- 
pressed in  such  manner  that  its  volume  remains  constant,  the  crystal  tends  to 
elongate  in  the  direction  of  its  principal  axis,  and  to  contract  in  a  direction 
perpendicular  to  this.  It  is  certain  that  even  if  there  is  no  change  of  volume 
there  is  a  change  of  shape,  and  hence  disgregation  work.  Even  if,  by  a  suit- 
able distribution  of  pressure  and  tension  upon  the  surface,  not  only  change  of 
volume  but  also  of  shape  is  prevented,  still  there  may  be  a  change  in  the  rela- 
tive direction  of  the  molecules,  if  not  in  the  relative  position  of  the  center  of 
gravity. 

This,  at  least,  seems  extremely  probable  from  the  change  of  optical  proper- 
ties in  different  directions,  caused  by  the  action  of  heat  upon  the  crystal,  and 
which  does  not  seem  accounted  for  by  simple  inequality  of  expansions. 

It  is  to  be  expected  that,  even  in  a  liquid,  every  change  of  temperature 
must  be  accompanied  by  a  perceptible  disgregation  work,  even  when  the  volume 
does  not  change,  when  the  liquid  is  near  the  point  of  solidification,  and  when, 
therefore,  the  lawless  arrangement  of  molecules,  characteristic  of  the  liquid 
condition,  tends  toward  a  regular  arrangement,  if  not  of  the  entire  mass,  at 
least  of  its  different  parts. 

We  see,  thus,  how  careful  we  must  be  before  we  assume  that  the  disgrega- 
tion work,  under  given  conditions,  is  zero.  The  invariability  of  the  mean 
distances  of  the  molecules  is  by  no  means  a  guaranty  of  this. 

Thus,  for  example,  water  cooled  below  the  temperature  corresponding  to 
its  maximum  density,  has,  for  the  same  pressure,  the  same  volume  at  two 
different  temperatures  four  degrees  apart.  The  outer  work  between  these  two 
conditions  is  zero,  but  nothing  justifies  us  in  assuming  the  disgregation  work 
as  zero  also.  We  can  scarcely  comprehend  the  anomaly  of  maximum  density 
otherwise  than  by  assuming  that  the  relative  direction  of  the  molecules  ap- 
proaches lavsr  and  definite  ar;  angement  the  more  the  freezing-point  is  approached, 
and  that,  when  for  two  different  temperatures  the  volume  is  the  same,  but  the 
arrangement  of  molecules  different,  the  transition  from  the  one  temperature  to 
the  other  is  accompanied  by  perceptible  disgregation  work. 


TO   THE  LECTURES.  65 


NOTE  7.— (Page  18.) 

AK  INCORRECT  DETERMIJSTATION  OF  THE   MECHAISTICAL   EQUIVALEIirT 
OF  HEAT. 

To  several  physicists,  among  them  Kupffer  and  Masson,  the  following  con- 
siderations have  appeared  permissible,  and  the  values  thus  obtained  for  the 
mechanical  equivalent  do  not  appear  to  deviate  much  from  the  true  value. 

Let  P  be  a  tension,  which,  when  applied  uniformly  to  the  surface  of  the 
unit  of  volume  of  a  body,  will  cause  an  expansion,  /I,  equal  to  that  caused  by 
a  rise  of  temperature  of  1  degree. 

The  work  of  this  force  which  must  be  applied  in  order  to  cause  the  exten- 
sion required  is  P^ 

On  the  other  hand,  we  must  impart  to  the  body,  in  order  to  make  it  expand 
through  the  same  distance,  a  quantity  of  heat  equal  to  the  product  of  the 
specific  heat  for  constant  pressure,  c^,,  into  the  weight  of  the  unit  of  volume, 
that  is,  the  density  D. 

If  the  work  PJ  were  the  mechanical  equivalent  of  this  amount  of  heat, 
we  should  have  the  relation 

P/i  =  Jc,D, 

which,  according  to  Kupffer,  is  confirmed  by  experiment* 

But  little  attention  is  required  to  comprehend  in  Avhat  respect  this  method 
of  treatment  is  defective.  The  heat,  Cj-D,  consists  of  three  parts  ■  1,  The 
increase  in  the  sum  of  the  living  forces  ;  2,  the  mechanical  equivalent  of  the 
disgregation  work  ;  3,  the  equivalent  of  the  outer  work. 

This  third  part  would  be  zero  if  the  expansion  took  place  in  a  vacuum. 
Under  ordinary  conditions  it  takes  place  under  atmospheric  pressure,  and  hence 
this  third  part  can  be  neglected  as  very  small  compared  to  the  second. 

It  is  essentially  different  with  the  first  part.  This  we  cannot  neglect  with- 
out implicitly  assuming  that  the  specific  heat  for  constant  volume  is  insig- 
nificant in  comparison  to  that  for  constant  pressure.  We  cannot,  therefore, 
regard  the  disgregation  work  as  the  mechanical  equivalent  of  the  entire  quan- 
tity CpD.  It  is,  moreover,  very  doubtful  whether  the  expression  P//  is  the 
precise  value  of  the  disgregation  work,  for  Pz/  is  the  work  of  the  forces  which, 
hj  their  mechanical  action,  cause  an  expansion  of  /i,  under  the  assumption 
that  the  temperature  of  the  body  remains  constavt.  If,  also,  there  is  nowhere  a 
development  of  perceptible  velocity,  it  is  only  under  similar  circumstances  the 
equivalent  of  the  disgregation  work.  Nothing  justifies  us  in  putting  this  work 
equal  to  that  in  the  body,  when,  by  the  action  of  heat,  it  expands  under  change 
of  temperature.  These  two  works  are,  indeed,  of  the  same  character,  and 
change  in  the  same  way,  when  we  pass  from  one  body  to  another,  but  it  is  at 
least  doubtful  that  they  are  identical. 

All  that  we  can  say  generally,  is,  that  the  resistance  to  tension  is  a  certain 
indication  of  the  intensity  of  the  molecular  forces,  and  that  a  considerable 
part  of  the  heat  which  is  imparted  to  a  body  is  employed  in  overcoming  these 
forces.  The  specific  heat  and  the  resistance  to  tension,  or  the  coefficient  of 
elasticity  which  measures  it,  change  in  the  same  way,  for  bodies  of  the  same 
kind,  as  metals. 

The  same  rather  superficial  law  may  be  extended  also  to  the  latent  heat  of 
liquefaction,  and  thus  it  happens  that  Person  has  been  led  to  find  a  numerical 

*  This  is  about  the  way  in  which  Masson  gives  the  view  of  Kupffer  in  his  Treatise  "Ueber 
die  Beziehung  der  physikalischen  Eigenschaften  der  Korper"  (Ann.  de  Chim.  et  de  Phys., 
3  serie,  vol.  liii.,  p.  256).  It  is  probable  that  this  interpretation  of  the  idea  of  the  learned 
Director  of  the  Physical  Observatory  at  St.  Petersburg  is  correct,  but  we  cannot  answer  for  it,  as 
in  the  original  text,  instend  of  the  i?harp  and  clear  expression  "work,"  we  find  always  the 
words  "  mechanical  effect,''  which  have  no  definite  signification  in  the  usual  vocabulary  of 
mathematics.  (Bulletin  de  la  classe  des  sciences  physiques  et  mathematiques  de  I'Academie  de 
St.  Petersbourg,  vol.  x.,  and  Pogg.  Annaleii,  vol.  Ixxxvi.,  p.  310.) 


66  NOTES  AND  ADDITIONS 

relation  between  tlie  coefficient  of  elasticity  and  the  liquefaction  heat  of  various 
metals,  which  may  be  regarded  as  approximately  indicated  by  experiment.  It 
is,  however,  impossible  to  deduce  any  such  relation  in  strict  accordance  with 
theory. 

It  is  probable  that  the  formula  of  Kupffer  has  about  the  same  value  as  that 
of  Person,  and  that  it  is  the  approximate  expression  of  a  relation  which  theory 
is  incapable  of  deducing. 

We  have  indeed  not  proved  that  this  formula  is  false,  but  simply  that  it 
cannot  be  deduced  from  any  a  priori  considerations.  If  Ave  consider  the  gen- 
eral fact  in  the  special  sense  that  the  coefficient  of  elasticity  and  the  specific 
heat  changed  together,  it  is  just  as  allowable,  and  about  as  valuable,  to  seek 
this  relation  by  experiment,  as  to  find  any  other  relations  by  comparison  with 
other  cases. 

Certainly,  such  a  comparison  can  have  no  such  legitimate  value  as  Kupffer 
attributes  to  it.  In  order  to  give  the  weight  P  as  a  function  of  the  coefficient 
of  elasticity,  Kupffer  uses  an  old  formula  of  Poisson,  which  everyone  now  rec- 
ognizes as  inexact,  and  indeed,  most  probably,  not  similarly  inexact  for  all 
bodies.  Hence,  a  factor  which  Kupffer  regards  in  his  calculations  as  constant, 
varies  for  different  metals,  and  since  this  change  is  not  yet  known  for  the 
metals  Kupffer  has  considered,  it  is  not  possible  to  introduce  the  necessary 
corrections  and  subject  the  empirical  value  of  his  formula  to  rigid  test. 


NOTE  8.— (Page  20.) 

BODIES   WHICH    CONTRACT   WHEN"   HEATED. 

It  is  almost  unnecessary  to  remark,  that  in  such  exceptional  cases  as  the 
melting  of  ice  and  change  of  volume  of  water  below  4",  in  wliich  there  is  a 
diminution  of  volume  under  the  action  of  heat,  the  discussion  is  reversed. 
We  consider  a  period  during  which  the  body  expands  while  cooling,  and  hence 
performs  outer  work  L  while  giving  up  heat  Q.  During  another  period,  let 
the  body,  by  the  application  of  outer  work  L ,  and  while  receiving  heat  Q' ,  be 
brought  back  to  its  original  condition.  If  L'  is  less  than  L,  we  obtain  an 
outer  work  L—L',  for  which  there  must  be  an  equivalent  absorption  of  heat  ; 
^'  must  be  greater  than  Q,  and  we  have 

L-L  =J{q-q). 

The  case  of  bodies  which,  within  certain  limits  of  temperature,  contract 
under  the  action  of  heat,  is  well  suited  to  direct  attention  to  the  views  given 
in  the  preceding  note.  If  we  limit  the  comparison  of  the  outer  work  with  that 
heat  which,  in  the  one  and  the  same  transformation,  is  imparted  or  abstracted 
from  the  body  in  order  to  change  it  from  one  condition  to  another,  we  are  led 
to  the  peculiar  conclusion  chat  the  geueratiou  of  heat,  as  well  as  its  disappear- 
ance, can  give  rise  to  work. 

Nothing  is  more  suited  to  make  apparent  how  necessary  it  is  to  take  into 
account  the  work  of  the  molecular  forces.  If,  by  some  local  disturbance,  by 
contact  with  a  piece  of  ice,  or  even  by  a  particle  of  dust,  we  cause  a  mass  of 
water  at  zero  to  crystallize,  the  molecular  forces  thus  called  into  play  by  this 
accidental  disturbance  of  equilibrium  place  the  molecules  in  those  positions 
which  constitute  a  solid  body,  and  the  positive  work  during  this  process  of 
change  has  for  its  equivalent  both  the  heat  generated  and  the  outer  work  per- 
formed by  expansion.  If,  inversely,  we  melt  the  ice,  the  heat  imparted  must 
be  the  equivalent  of  the  excess  of  the  disgregation  work  over  the  outer.  In 
ordinary  cases,  on  the  other  hand,  the  heat  imparted  during  melting  and  with- 
drawn during  solidification  is  the  equivalent  of  the  sum,  and  not  of  the  differ- 
ence of  the  disgregation  work  and  outer  work. 

If  a  strip  of  vulcanized  rubber  is  elongated  by  tension,  we  have  a  rise  of 


TO  THE  LECTURES.  67 

temperature,  while  the  temperature  of  a  metal  wire,  under  the  same  circum- 
stances, is  lowered.  This  is  due  to  the  fact  that  heat  expands  the  metal,  but 
contracts  the  rubber.     This  is  a  point  which  Joule  has  completely  cleared  up.* 


NOTE  9.— (Page  20.) 

UPON   CALOEIMETRIC   MEASUREMENTS   IN   WHICH   NO   ACCOUNT    HAS 
BEEN   TAKEN   OF   THE   OUTER   WORK. 

The  necessity  of  taking  account,  in  all  phenomena  depending  upon  the 
action  of  heat,  of  the  outer  work,  would  seem  to  justify  the  fear  that  a  large 
part  of  our  calorimetric  measurements  are  liable  to  be  affected  by  a  fundamen- 
tal error,  as  they  were  made  at  a  time  when  the  principle  of  the  mechanical 
theory  of  heat  was  scarcely  suspected.  A  little  consideration,  however,  will 
serve  to  show  that  such  fear  is  groundless.  Strictly  speaking,  we  must  un- 
doubtedly admit  that  specific  heat  and  latent  heat  depend  always  upon  the 
outer  pressure  under  which  bodies  expand  or  change  their  aggregate  condition. 
But,  under  ordinary  circumstances,  the  outer  work  is  so  small  for  solids  and 
liquids,  that  such  dependence  gives  rise  to  a  correction  so  slight  that  it  is  im- 
perceptible even  to  the  most  sensitive  methods  of  measurement. 

For  gases,  the  influence  of  such  correction  is  so  great  that  account  has 
always  been  taken  of  it,  and  it  has  always  been  held  as  indispensable  to  give, 
for  example,  the  pressure  of  a  gas  under  which  the  specific  heat  has  been  de- 
termined.    Only  in  the  case  of  vapors  have  errors  been  committed. 

Every  investigation  upon  the  latent  heat  of  vaporization,  in  which  an  outer 
work  is  not  performed  upon  the  steam  when  condensing  equal  to  that  per- 
formed by  it  during  its  formation,  is  essentially  erroneous,  and  can  give  no 
reliable  result. 

Eegnault  has,  therefore,  very  properly,  in  his  experiments  upon  the  latent 
heat  of  vaporization  of  water,  maintained  in  all  parts  of  his  apparatus  a  uni- 
form pressure.  The  new  theory  by  no  means  invalidates  the  value  of  the 
results  obtained  by  this  distinguished  physicist,  but  it  rather  adds  to  their 
weight,  and  uses  them  to  attain  new  results.  It  does,  however,  deprive  numer- 
ous investigations,  in  whicli  this  precaution  is  neglected,  of  all  claim  to  relia- 
bilitv. 


NOTE  10.— (Page  21.) 

THEORY    OF   THE    CONSTITUTION    OF   GASES. 

If  we  suppose,  in  a  confined  space,  a  large  number  of  molecules  separated 
by  such  intervals  that  their  mutual  actions  may  be  neglected,  and  assume  that 
these  molecules  are  at  rest,  it  is  evident  that  they  can  exert  no  influence  on 
each  other,  and  that  a  portion  of  these  molecules  can  undergo  any  change  of 
state,  without  affecting  in  the  least  the  condition  of  the  others.  Upon  bodies 
which  confine  the  system,  there  can  be  no  such  action  as  pressure.  Individual 
molecules  may  indeed  be  so  near  the  bounding  body  as  to  act  upon  it ;  but,  by 

*  Joule,  Phil.  Mag.,  vol.  xvi.,  p.  227.  1857;  Ann.  de  Ch:m.  et  de  Phys.,  vol.  Hi.,  p.  226; 
Thomson,  Phil.  Mag.,  1857,  vol.  viii ,  p.  504.  See  also  Tyndall,  Heat  as  a  Mode  of  Motion,  2d 
Ed.,  p.  11.5;  also  Villari,  Pogg.  Ann.,  vol.  cxliv.,  p.  274;  Schmulewitsch,  Vierteljahvesschiift 
der  naturforsch.  Gesellscliaft  in  Ziirich.  Jahrgang  xi.,  Heft  3,  and  Pogg.  Ann.,  vol.  cxliv., 
p.  280. 

This  propertj'  was  first  discovered  by  Gough.  with  non-vulcanized  rubber,  in  1800.  Nicholson's 
Journ..  vol.  xiii.,  p.  805;  Gehlen's  Journ.,  vol.  ix.,  p.  217. 

Later  Reusch  observed  similar  phenomena  with  gutta-percha.  Pogc.  Ann,,  vol.  cxliv., 
p.  315. 

Govi  ascribes  the' heating  under  tension  to  numberless  gas  bubbles. 


68  NOTES  AND  ADDITIONS 

reason  of  the  assumption  of  llie  mean  distance  of  the  molecules  apart,  the 
number  will  be  very  small  in  comparison  to  the  number  of  molecules  which 
must  act  together  in  order  to  caase  the  pressure  of  a  liquid  upon  a  solid  or 
upon  another  liquid. 

Certainly,  nothing  can  resemble  less  a  gas  than  this  incoherent  collection, 
which  can  hardly  be  called  even  a  system.  We  have,  nevertheless,  seen  in  the 
text  that  it  is  not  easy  to  do  without  the  assumption  that  in  gases  the  distances 
apart  of  the  molecules  is  incomparably  greater  than  for  any  other  bodies.  If, 
however,  we  ascribe  motion  to  these  molecules,  the  state  of  things  is  changed, 
and  the  known  properties  of  a  perfect  gas  are  necessary  consequences  of  such 
an  assumption. 

In  consequence  of  their  motion,  the  molecules  will  impinge  upon  each  other 
and  upon  the  bounding  surface.  In  a  short  time  there  will  be  a  mean  condi- 
tion, the  chief  properties  of  which  can  be  easily  recognized.  By  reason  of  the 
size  of  the  inter-molecular  spaces,  almost  all  the  molecules  must  move  at  any 
moment,  as  if  influenced  by  no  deviatinsr  forces  ;  that  is,  in  straight  lines,  and 
with  a  uniform  velocity  common  to  all  the  molecules  in  the  final  condition,  but 
different  for  different  molecules.  Those  which  accidentally  approach  each 
other  at  any  moment  act  upon  each  other,  and  mutually  influence  each  other's 
paths  and  velocities.  But  these  changes  last  but  a  short  time,  after  whicli  the 
molecules  recede  and  return  to  the  general  conditions  of  the  system.  Individ 
ual  molecules  may  also  impinge  centrally  or  obliquely  ;  but,  since  both  the 
masses  and  velocities  of  individual  molecules  are  by  hypothesis  equal,  the  di- 
rection of  the  velocities  may  be  altered  by  impact,  but  not  their  amounts.  We 
see,  therefore,  that,  in  order  to  find  the  action  which  the  system  exerts  upon 
the  confining  boundaries,  we  can  assume  as  the  actual  condition  one  in  which 
all  the  molecules  move  incessantly  in  straight  lines  in  all  conceivable  directions 
without  striking. 

If  the  boundaries  are  perfectly  elastic,  every  impinging  molecule  will  be 
thrown  back,  the  direction  of  its  motion  changed,  but  its  velocity  unchanged, 
so  that  the  total  condition  of  the  system  remains  invariable.  Let  us  assume 
this  condition  as  fulfilled,  and  seek  what  force  must  be  exerted  upon  a  bound- 
ary of  given  surface  ;  what  pressure,  for  example,  must  be  applied,  in  order 
to  keep  it  immovable.  This  force  must  be  capable  of  reversing  the  normal 
component  of  the  velocity  of  every  molecule  impinging  in  a  given  time,  or, 
what  amounts  to  the  same  thing,  of  imparting  a  normal  velocity,  in  the  opposite 
direction,  of  double  the  intensity  of  the  component. 

This  must  evidently  be  proportional  to  the  uniform  velocity  of  the  mole- 
cules and  ttieir  masses.  It  must  also  be  proportional  to  the  number  of  molecules 
impinging  in  a  given  time,  that  is,  to  the  number  of  molecules  in  a  unit  of 
volume,  and,  further,  proportional  a  second  time  to  the  velocity  ;  for  the  time 
which  any  molecule  requires  to  traverse  the  space  between  two  boundaries  is 
inversely  proportional  to  the  velocity,  and  hence  the  number  of  impacts  which 
any  molecule  makes  in  a  given  time  is  proportional  to  the  velocity. 

The  pressure,  therefore,  which  must  be  exerted  is  proportional  to  the  mass 
and  number  of  molecules  in  the  unit  of  volume,  and  to  the  square  of  the  ve- 
locity. 

The  proportionality  between  pressure  and  mimber  of  molecules  is  nothing 
more  than  the  proportionality  between  jiressure  and  density  expr-essed  by 
Mariotte's  law. 

The  proportionality  to  the  mass  and  square  of  the  velocity  is  also  easily 
interpreted.  « 

If  we  accept  the  usual  views  as  to  the  nature  of  heat,  we  can  regard  the  veloc- 
ity of  the  molecules  as  an  indication  of  the  temperature  of  the  gas,  v/liich  changes 
in  the  same  degree  as  the  temperature  itself.  We  have  thus  a  theoretical  de- 
finition of  equality  of  temperature.  We  say  that  two  gases  possess  the  same 
temperatures  if,  when  united  under  the  same  jiressure,  they  do  not  affect  each 
other's  condition.  If  we  assume  that,  for  two  different  gases  under  the  same 
conditions,  there  are  in  equal  volumes  an  equal  number  of  molecules,  then  the 
temperatures  are  alike  when  the  product  of  the  mass  of  a  molecule  by  the 
square  of  the  velocity  is  the  same  in  both. 


TO   THE  LECTURES.  69 

The  equality  of  the  living  force  of  the  molecules  includes,  therefore,  the 
equality  of  temperature. 

In  other  words  we  may  say,  that  the  living  force  of  the  molecules  is  a  func- 
tion of  the  temperature,  which  is  the  same  for  all  gases.  The  proportionality 
of  the  pressure  with  the  living  force  indicates,  therefore,  that  in  all  gases  the 
relation  between  pressure  and  temperature  is  the  same.  From  this  identity, 
together  with  Mariotte's  law,  we  easily  deduce  the  agreement  in  coefficient  of 
expansion.  If,  as  is  customary,  we  measure  the  temperature  by  the  air-ther- 
mometer, we  know  that,  if  the  temperature  indicated  is  t,  and  the  coefficient  of 
expansion  is  a,  the  pressure  for  constant  volume  is  proportional  to  the  expres- 
sion 

-  -t  t        or        373  +  t. 
a 

The  living  force  of  the  molecules  is,  therefore,  proportional  to  the  tem- 
perature measured  by  a  thermometer  for  which  the  zero  point  is  at  —  273  C. 
At  this  temperature  of  —  273  the  living  force  of  the  molecules  is  zero,  or  we 
may  say  that  at  this  temperature  the  gas  contains  no  uiore  heat — the  absolute 
zero  of  temperature  is  then  reached,  and  the  gas  ceases  to  be  a  gas,  and  becomes 
that  incoherent  mass  of  atoms,  independent  and  immovable,  which  we  have  just 
spoken  of. 

If,  finally,  we  assume,  with  all  chemists,  that,  under  the  same  pressure,  all 
simple  gases  contain  in  equal  volumes  the  same  number  of  molecules,  the 
changes  of  temperature  are  proportional  to  the  changes  in  the  living  force  of 
each  molecule.  We  see  then,  that  in  order  to  heat  equal  volumes  of  different 
gases  the  same  number  of  degrees,  the  same  amount  of  heat  is  necessary.  This 
conclusion  is  direct  when  rise  of  temperature  occurs  without  change  of  volume  ; 
it  follows,  also,  when  there  is  a  change  of  volume,  when  we  consider  the  for- 
mulse  on  page  23. 

Thus,  the  characteristic  properties  of  perfect  gases  find  a  simple  and  natural 
explanation.  The  idea  of  a  "perfect  gas"  is  itself  sharply  defined,  and  it  is 
easy  to  conceive  of  imperfect  gases,  which  do  not  strictly  follow  Mariotte's  law, 
whose  coefficient  of  expansion  varies  with  the  pressure,  and  which,  for  equal 
volumes,  do  not  possess  the  same  heat  capacity,  such  as  air  and  oxygen.  In 
the  system  of  individual  molecules  moving  rapidly  in  all  directions,  which  we 
have  considered,  we  have  assumed  that  at  any  given  moment  the  number  of 
molecules,  whose  motion  is  not  rectilinear  and  uniform,  is  inconsiderable  in 
comparison  with  those  whose  motion  satisfies  these  two  conditions  ;  or,  what 
amounts  to  the  same  thing,  that  for  each  molecule  the  duration  of  the  period 
of  disturbance  is  vanishingly  small  compared  to  the  period  in  which  the  motion 
is  uniform.  If,  now,  the  ratio  of  these  two  periods,  while  indeed  still  very 
small,  is  not  venishingly  so,  the  preceding  considerations  do  not  hold  strictly; 
and  our  conclusions  no  longer  represent  with  precision  the  properties  of  the 
system,  but  are  more  or  less  approximate  expressions  of  these  properties.  It 
is  also  evident  that  the  more  we  diminish  the  distances  apart  of  the  molecules, 
that  is,  the  more  we  condense  the  gas,  the  less  reason  we  have  to  presume  per- 
fectly uniform  motions,  and  the  more,  therefore,  is  the  deviation  from  the  con- 
dition of  a  perfect  gas.  This  perfect  condition  is,  in  fact,  an  ideal  state  toward 
which  gases  approach  as  their  state  of  rarefaction  increases,  but  which  they 
can  never  exactly  attain.* 

*  The  theory  given  in  this  note  is  by  no  means  new.  It  wa*  indicated  by  Daniel  Bernoulli  in 
his  Hj'drodjTiamics.  m  17.38. 

After  being  forgotten  by  the  world  it  was  revived  again,  about  1822,  by  Herapath.  Only  in 
recent  times  has  it  received  its  present  shnpe  by  Joule.  Kronig.  and  Clausius.  Clansius  has 
treated  it  in  the  most  general  manner,  and  added  to  its  completeness  by  taking  into  account,  in 
addition  to  the  rectilinear  motion  of  the  molecules,  their  inner  motions,  motions  of  rotation,  and 
the  probable  motions  of  imponderable  liiiids.  In  a  presentation  designed  to  be  elementary  in 
character,  no  account  can  be  taken  of  such  a  treatment. 

It  may  snftice  to  refer  for  further  information  to  the  original  treatises  of  Joule,  KrOnig,  and 
Clausius.  Joule,  Phil.  Mag.  4  Ser.,  vol.  xiv.,  p.  211 ;  Kronig,  Fogg.  Ann.  vol.  xcix.,  p.  315  : 
CiauBiirs,  Pogg.  Ann.  vol.  c.,  p.  353  ;  and  Abhandlungen,  vol.  ii.,  p.  229. 


70  NOTES  AND  ADDITIONS 

NOTE   11.— (Page  23.) 

HOW  GASES   AND   VAPORS  PEEFORM   OUTER   WORK. 

The  theory  given  in  Note  10  does  not  deduce  the  pressure  of  gases  from 
the  direct  action  of  a  repulsive  force,  but  refers  it  to  incessant  impacts.  It  is 
thus  possible  to  conceive  how  a  change  of  volume  can  be  accomjianied  by  no 
inner  work,  although  all  gases  seem  to  possess  the  tendency  to  expand  and 
to  resist  compression.  When  the  volume  of  a  gas  changes,  the  number  of  mole- 
cules in  a  given  space  changes  also,  and  when  a  change  of  temperature  occurs, 
the  velocity  of  the  molecules  changes.  But  so  long  as  the  mean  interval 
between  the  molecules  does  not  exceed  certain  limits,  their  mutual  actions  are 
imperceptible,  as  well  after  as  before  the  change  of  volume,  and  hence  require 
no  work.  The  mechanism  of  the  relation  which  exists  between  outer  work 
and  heat  absorbed  or  generated,  is  not  hard  to  conceive.  When  we  compress 
a  gas,  we  apply  a  force  to  a  movable  piston  which  is  greater  than  that  neces- 
sary to  reverse  all  the  noi*mal  components  of  all  the  molecules  which  in  a  given 
time  impinge  upon  the  piston. 

The  velocity  of  all  the  molecules  is  thus  directly  or  indirectly  increased, 
and  the  work  of  the  outer  pressure  is  equivalent  to  the  increase  in  the  sum  of 
the  living  forces  of  the  molecules,  that  is,  to  the  heat  generated. 

The  reverse  is  the  case  for  expansion.  The  molecules  impart  to  the  piston, 
upon  which  there  is  no  sufficient  force,  according  to  the  laws  of  impact,  a 
portion  of  their  living  force,  and  this  impartation  of  living  force  is,  from  our 
standpoint,  an  absorption  of  heat  or  a  production  of  work. 

Similar  considerations  apply  to  vapors  and  their  work  in  the  engines  they 
operate.  There  is  thus  an  imparting  of  living  force  when  the  piston  is  raised, 
and  a  reappearance  of  living  force  when  it  sinks.  If  work  is  performed  by 
the  engine,  it  is  sufficient  that  there  is  no  equality  in  the  two  cases.  The 
living  force  which  fails  to  reappear  is  the  equivalent  of  the  work  done.  Thus 
disappears  the  apparent  contradiction  in  the  fact  that  a  system  may  produce 
work,  while  the  inner  works  are  zero. 


.  NOTE   12.— (Page  24.) 

THE     VALUE     OF     THE     MECHANICAL     EQUIVALENT     OF     HEAT      AS 
GIVEN    BY    CARBONIC    ACID. 

In  determining  the  mechanical  equivalent  of  heat  from  the  properties  of 
carbonic  acid,  the  specific  heat  at  constant  pressure  was  taken  at  0.2163,  as 
given  by  Regnault  in  April,  1853.  According  as  we  take  for  the  ratio  of  the 
two  specific  heats,  1.2867  as  given  by  Masson,  or  1.3382,  given  by  Dulong,  we 
obtain  402  or  355.  But  the  number  0.2163  expresses  only  the  mean  specific 
heat  between  0  and  210 'C,  and  this  mean  specific  heat  is  not  the  exact  value 
for  a  given  temperature. 

From  the  experiments  of  Regnault  (given  in  Vol.  26  of  the  Memoires  de 
I'Acadeinie  des  Sciences),  carbonic  acid  has  at  the  temperatures  0  and  100  the 
specific  heat  0.1870  and  0.2645.  If  we  insert  these  values  in  the  formulae  on 
page  23,  we  have  for  /the  values 

410  and  357,  or  465  and  406, 

ri 
according  as  we  take  for  -^  the  value  given  by  Masson  or  by  Dulong. 


TO  THE  LECTURES.  7I 


NOTE   13.— (Page  25.) 

PRINCIPLE    OF   THE    METHOD    OF    USTVESTIGATIOlSr    OF   THOMSON"   AND 
JOULE   OF   THE   HEAT   PHENOMENA   IN    GASES   IN   MOTION. 

The  method  of  investigation  used  by  William  Thomson  and  Joule  consists 
in  allowing  a  stream  of  gas  to  pass  through  a  porous  diaphragm,  from  which 
it  issues  with  considerably  reduced  pressure.  The  friction  absorbs  nearly  ail 
the  velocity  due  to  the  expansion.  A  sensitive  thermometer  gives  the  tem- 
perature of  the  gas  before  and  after  efflux. 

It  was  so  arranged  for  air,  carbonic  acid,  and  hydrogen,  that  simple  expan 
sion,  even  when  accompanied  by  no  outer  work,  caused  a  slight  change  of 
temperature,  nearly  proportional  to  the  pressure,  and  which  depended  on  the 
initial  temperature.  Thus  has  been  determined  the  relation  between  the  inner 
work  and  the  outer,  when  a  gas  expands  under  pressure.  If  we  assume  the 
expansion  as  very  small,  and  if  the  temperature  is  nearly  15%  the  ratio  cf 
these  values  for  air.  is  -477 ;  for  carbonic  acid,  7^ ;  for  hydrogen,  completely  im- 
perceptible. 

The  formula  of  page  23,  which  gives  the  mechanical  equivalent  of  heat 
from  the  work  performed  by  slight  expansion  and  the  heat  absorbed,  is  there- 
fore applicable  without  error  to  hydrogen.  For  air  there  is  a  slight  error, 
which,  however,  is  less  than  the  error  which  may  ajise  from  the  inaccuracy  of 
the  value  for  the  specific  heat  for  constant  volume.  For  carbonic  acid,  finally, 
the  left  side  of  the  equation  must  be  increased  lay  TStli.* 

It  is  certainly  too  early  to  seek  in  this  manner  to  obtain  a  satisfactory  agree- 
ment in  the  values  of  the  mechanical  equivalent  of  heat  for  various  gases. 

The  density,  the  coefficient  of  expansion,  the  specific  heat  for  constant  volume, 
are  very  exactly  known  from  Kegnault's  experiments,  for  air,  hydrogen,  and 
carbonic  acid.  But  there  &til)  remains  considerable  uncertainty  in  the  values 
attributed  to  the  specific  heats  for  constant  volume.  This  evades  direct  deter- 
mination, and  must  be  deduced  from  observations  upon  the  velocity  of  sound, 
or  from  heat  phenomena  which  are  caused  by  changes  of  volume;  and,  in  the 
present  state  of  such  experiments,  we  can  scarcely  assume  that,  except  in  the 
case  of  air,  it  has  been  determined  with  accuracy.  It  follows,  moreover,  from 
the  formula  and  the  known  values  of  CV  and  G,-,  that  every  error  which  is 
made  in  the  value  of  6V,  in  the  case  of  air,  causes  a  double  error  in  the 
resulting  value  of  /,  and,  in  the  case  of  carbonic  acid,  a  threefold  error.f 

*  In  ii  treatise  e.=pecially  iiiteiicled  to  diminish  the  difEerence  between  the  values  of  the 
mechanical  equivalent  as  given  by  the  forniula,  Baum<iartner  has  taken  the  ratio  of  the  inner 
to  the  outer  work,  according  to  Thomson  and  Joule,  equal  to  -i-^  for  hydrogen,  -  j^  for  air,  and 
-.i-_^  for  carbonic  acid. 

These  values  are  certainly  given  by  Thomson  and  Joule,  but  ihey  relate  to  the  case  in 
which  the  pressure  fails  from  4.7  atmospheres  to  one  atmosphere,  It  is  a  great  error  to  apply 
them  as  corrections  to  a  formula  deduced  for  such  a  slight  change  of  pressure  as  thatwhicli 
accompanies  a  change  of  volume  of  the  amount  of  the  coefficient  of  expansion.  (Sitzungs-be- 
vichte  der  K.  K.  Akademie  der  Wissenschaften,  Vienna,  vol.  xxxviii.,  p.  344.) 

t  If  we  assume  the  density  of  the  air  about  -j^,^,yj,  its  specific  heat  under  constant  pressure 
to  be  known,  its  coefficient  of  expansion  at  about  i^}-^-,,  and  the  ratio  of  the  specific  heats  at  about 
-l.j,  we  find  in  the  value  of  ./  deduced  from  the  formula  an  error  of  ^i-,  or  about  8  units. 

For  carbonic  acid  the  difEerence  m  the  value  of  C''.  given  by  the  experiments  of  Dulong  and 
Masson,  is  so  great  that  no  dependence  can  lie  placed  upon  calculation.  We  see,  therefore,  that 
it  IS  not  yet  tiine  to  discuss  this  correction,  and  all  that  we  can  say  with  safety  is,  that  the  coin- 
cidence of  the  results  for  air  and  hydrogen  renders  it  certain  that  the  value  of  the  equivalent 
must  lie  between  420  and  430. 


72  NOTES  AND  ADDITIONS 

NOTE   14.— (Page  32.) 

UPOISr   THE    COKDEKSATIOJSr   OF   STEAM   IN   EXPANDING. 

The  condensation  of  saturated  steam  during  expansion  was  shown  on  theo- 
retical grounds  by  Rankine  and  Clausius  independently  in  1850.  Theory  gives 
a  necessary  relation  between  the  latent  heat  of  vaporization  of  water,  its  spe- 
cific heat,  and  the  amount  of  heat  which  must  be  imparted  to  the  unit  of  weight 
of  steam,  when  simultaneously  heated  and  compressed,  so  that  it  shall  remain 
saturated.  Since  all  the  quautities  which  enter  the  equation,  with  the  excep- 
tion of  the  third,  are  given  by  Regnault's  experiments,  this  may  be  found,  and 
in  this  way  it  has  been  found  to  be  negative.  We  must,  therefore,  abstract 
heat  from  steam  which  is  compressed  and  heated,  in  order  to  keep  it  saturated  ; 
and,  on  the  other  hand,  heat  must  be  imparted  to  steam  which  expands  and  cools, 
in  order  to  keep  it  from  condensing. 

When  the  expansion  occurs  without  addition  of  heat  from  without,  all  the 
steam  cannot  therefore  remain  in  the  pure  saturated  state,  and  in  order  that  a 
part  of  it  may  remain  saturated,  another  part  must  condense,  and  thus  furnish 
the  necessary  heat. 


NOTj:  15.— (Page  34.) 

THE   EEGEifEEATOR   Il^T   HOT-AIR   EISTGIN'ES. 

It  may  appear  as  if  the  same  reasons  which  cause  the  heat  quantity  q'  to  be 
lost  to  the  engine,  also  contradicted  the  possibility  of  the  unlimited  usefulness 
of  the  quantity  c„  {t^  —  tn). 

In  fact,  we  can  scarcely  see  any  other  way  of  reducing  the  temperature  from 
ij  to  tn,  than  by  contact  with  a  cold  body,  which  receives  heat  as  the  gas  cools, 
but  which,  at  the  end,  has  the  same  temperature—^,, — as  the  gas.  Under  such 
circumstances,  indeed,  the  quantity  of  heat  c-it^  —  t„)  would  be  contained  in  a 
body  of  the  temperature  to,  and  could  not  therefore  be  used  for  heating  a  new 
charge  of  gas,  so  that  it  would  also  be  lost  as  well  as  the  heat  q'.  This  diffi- 
culty has  been  met  by  Stirling  in  a  very  ingenious  manner.  The  gas  is  cooled 
in  theengine  from  ^,  to  to,  by  passing  thi'ough  a  porous  conducting  body,  such 
as  a  net-work  of  wires,  to  the  different  layers  of  which  it  imparts  its  heat.  If 
this  body  is  at  first  at  the  temperature  to,  its  different  layers  will  be  raised  by 
contact  with  the  departing  gas  to  some  temperature  higher  than  to,  but  some- 
what less  than  ti,  except  the  last  layer,  v,'hich,if  the  body  has  sufficient  thick- 
ness, will  have  the  temperature  to-  When,  then,  a  second  charge  of  gas  of 
the  temperature  to  enters,  it  will  be  gradually  heated  by  contact  with  the  suc- 
cessive layers,  and  will  enter  the  cylinder  with  a  higher  temperature  than  to, 
so  that  in  order  to  raise  this  charge  up  to  ^,  will  require  less  heat  than  the  first 
charge. 

After  this  charge  has  acted  in  the  engine,  and  passes  out,  it  finds  all  the 
layers,  with  the  exception  of  the  last,  at  a  higher  temperature  than  to,  and  will 
raise  them  all  to  a  higher  temperature  than  the  first  did.  Thus,  a  third  charge 
will  enter  the  cylinder  hotter  than  the  second,  and  upon  issuing,  will  leave  the 
layers  still  hotter  than  the  second  did,  and  the  temperature  of  the  first  layer 
still  nearer  therefore  the  temperature  t-i  of  the  issuing  charge.  The  amount 
of  heat  which  must  be  imparted  by  the  fire  in  order  to  raise  each  successive 
charge  of  air  from  ^n  to  ^°i,  will  decrease. 

Theoretically,  the  engine  approaches  constantly  the  condition  in  the  text, 
in  which  the  heat  c,-  {t^  —  to)  is  constantly  given  up  by  the  air  to  the  regenera- 


TO   THE  LECTURES. 


73 


tor,  and  then  given  back  by  tbe  latter  to  the  fresb  cbarge,  thus  remaining 
always  in  the  engine. 

In  practice,  a  certain  fraction  of  this  heat  must  be  lost  and  made  good  by 
the  fire  at  every  stroke.  Experiment  shows  that  the  heat  of  this  fraction  may 
be  less  than  /lyth. 

The  porous  body  which  is  thus  used  is  called  the  "  regenerator."  It  has 
been  constructed  in  various  ways.  A  system  of  glass  tubes  has  been  used,  also 
metal  wires,  and  wire  net-work.  Glass  and  similar  substances  are  too  bad 
conductors,  and  answer  the  purpose  very  imperfectly.  Metal  wires  and  net- 
work are  better,  but  they  are  quickly  oxidized  by  the  action  of  the  hot  air. 

This  purely  practical  difficulty  is  one  of  the  chief  hindrances  to  the  extended 
application  of  the  hot-air  engine. 


NOTE  16.— (Page  35.) 

DETEEMINATIOlSr     OF    THE   EFFICIENCY    OF   THE    EEICSSOIf' 
AND    THE   ENGINE   WITHOUT   EEGENEEATOE. 


Let  us  take  as  an  example  an  Ericsson  hot-air  engine.  In  this  the  air  is 
first  heated  under  constant  pressure,  then  allowed  to  expand  and  cool,  then  still 
further  cooled  under  constant  pressure,  and  finally  by  compression  brought  back 
to  its  original  condition.  As  in  Stirling's  engine,  we  can  represent  these  succes- 
sive operations  graphically.  Let  0  I'a  be  the  volume  Vq  of  the  unit  of  weight  of 
air,  of  the  temperature  tg;  and  pressure^,,.  Thus  Vq  Tq  is  the  pressure.  The 
air  is  first  heated  under  this  constant  pressure  po,  from  the  temperature  ^o  to 
the  temperature  t^,  which  re- 
quires the  amount  of  heat 
Cj.  {t-i^  —  tu).  Let  0  «!  be  the 
volume  ■Wi  at  the  end  of  this 
operation. 

Now  the  air  expands  from 
the  volume  Vi  to  Oi^.,  =  ^o, 
while  the  temperature  t-^  re- 
mains unchanged.  The  or- 
dinates  to  the  hyperbola 
Ti  T-i,  give  at  every  instant 
of  ex;pansion  the  pressure  of 
the  air.  Let  the  final  jjress- 
ure  be  j9,2.  Next,  the  air  is 
cooled  under  this  constant 
pressure  p.,,  until  the  tem- 
perature is  again  tc.  It  is 
finally  compressed,  while  the 
temperature  remains  un- 
changed, till  the  pressure  is 

again  p,,.  The  hy})erbola  To  Tq  gives  the  pressure  at  any  point  during  the 
compression.  The  area  I'n  T-^  T^  T,-,  is  evidently  the  geometrical  representa- 
tion of  the  outer  work.  We  can  easily  find  this  by  prolonging  the  lines  T^  Tq 
to  intersections  B  and  S  with  the  axis  of  Y,  and  finding  the  difference  of  the 
hyperbolic  areas  B  8  T^  1\  and  B  8  To  T^.     We  thus  find 


Y 

Ta                    T. 

S 

po 

Ti 

Pe 

0 

T 

(o                " 

l-^ 

'"3                           ■^ 

^z 

Pig.  5. 


Area       To  T^  T^  To  =  («i  -  Vo)Po  log  nat 


P2 


The  heat  utilized  is  therefore  equal  to  the  quotient  of  this  expression  when 
divided  by  the  mechanical  equivalent  of  heat.     As  to  the  total  heat  imparted 


74 


NOTES  AND  ADDITIONS 


and  the  heat  not  utilized  we  have,  when   we  denote  by  q  the  heat  imparted 
during  the  second  operation,  and  by  q'  that  abstracted  during  the  fourth, 

Cp{ti-tn)  +  q 
c,>{.tx-to)  +  q'. 

We  can,  therefore,  just  as  in  Stirling's  engine,  by  means  of  a  regenerator,  keep 
the  heat  Cp{ti  —  t(,)  in  the  engine. 

Finally,  the  amounts  of  heat  q  and  q'  are  the  heat  equivalents  of  the  work 
represented  by  the  hyperbolic  areas  v,  Ti  T,  V-i  and  Vq  Tq  T,j  -y,,.     These  are 

Pq  Vi  log  nat  — ,   and  Pq  Vq  log  nat  — -  . 
P2  Pa 

The  ratio  of  the  useful  work  to  the  total  is  hence 


or  again 


ajti  -to) 
1  +  at,     ' 


Let  us  consider,  finally,  a  third  kind  of  engine,  which,  indeed,  is  not  prac- 
tically realized,  but  which  is  theoretically   the  most  perfect,  since  it  does  not 
require  a  regenerator.     The  air  first  expands,  while   its  temperature  is  kept 
constantly  T^  by  the  addition  of  heat.     The  hyperbolic  arc  J',  T^  gives  the  re- 
lation between  pressure  and 
volume   during    the    expan- 
sion. 

Let  the  initial  pressure  be 
Pi  and  the  final  p. 2.  We  now 
let  the  air  still  further  ex- 
pand, but  without  adding 
or  abstracting  heat,  so  that 
the  temperature  gradually 
sinks  to  to,  and  the  press- 
ure varies  as  the  ordinates  to 
the  curve  T,  Tq,  which  must 
approach  the  axis  of  x  more 
rapidly  than  2^1  r,.  Let  p , 
be  the  final  pressure,  and  to 
the   corresponding   tempera- 

tX     ture.     In    the    third    period 

Ti    /o-  7-2     ''3  the  air  is  compressed,  while, 

Fig.  6.  at  the  same  time,  heat  is  ab- 

stracted in  such  a  manner 
that  the  temperature  is  kept  constant,  so  that  the  hyperbolic  arc  To  Tg  gives 
the  relation  between  pressure  and  volume.  This  compression  is  carried  up  to 
the  point  where  the  pressure  ispo.  Finally,  the  air  is  still  further  compressed, 
but  without  imparting  or  abstracting  heat,  until  the  pressure  is  again  ^j,  and 
the  temperature  t-^. 

We  see  at  once  the  analogy  with  the  preceding  cases.     In  the  first  operation 
the  heat  imparted  is 


Pi  '^1 


nat 


and  in  the  third,  that  abstracted  is 


1  1  ^  Po 

-jPo  Vo  log  nat  —  . 
«/  Pa 


TO   TEE  LECTURES.  75 

Moreover,  when  a  is  the  coefficient  of  expansion,  we  have,  for  the  relation  be- 
tween pressure  and  temperature  in  the  second  and  fourth  periods,* 


1  +  octo)  P-,  ' 


and 


(l  +  atA " 

U  +  ato] 


Hence,  ^^Zl,     or     ^  =  ^. 

Pi         Po  Ps         Pi 

We  have,  therefore,  for  the  ratio  of  the  heat  utilized  to  the  total  expenditure 
of  heat 


i?i  «i  1  +  oct^ 


N"OTE  17.— (Page  36.) 

HOT-AIR    ENGINES    IN    WHICH    THE    TEMPERATITEE    FALLS    TO    THE  , 
ABSOLUTE    ZEEO    OF    TEMPEEATUEE. 

It  follows  from  the  general  formula  that  if  it  were  possible,  in  a  hot-air 
engine  which  satisfies  the  above  conditions,  to  reduce  the  temperature  down 
to  the  absolute  zero,  the  efficiency  would  be  unity.  It  is  not  difficult  to  see 
the  reason.  In  Stirling's  engine,  for  example,  if  the  third  operation — viz.,  that 
in  which  the  gas  is  compressed  under  withdrawal  of  heat — takes  place  under 
the  temperature  of  absolute  zero,  the  gas  possesses  at  this  temperature  no 
pressure.  No  work,  therefore,  is  necessary  to  compress  it,  and  the  total  work 
performed  in  the  second  period  will  be  disposable.  In  Ericsson's  engine  the  gas, 
in  order  to  have,  at  a  temperature  indefinitely  near  the  absolute  zero,  a  notice- 
able pressure,  must  have  an  indefinitely  small  volume.  The  work  in  the  fourth 
operation  will  be  infinitely  small,  and  we  have  at  disposal  the  total  work  in  the 
second  period.  In  the  engine,  finally,  without  regenerator,  if  the  third  opera- 
tion, as  in  Stirling's  engine,  occurs  at  the  temperature  of  absolute  zero,  it  re- 
quires also  no  expenditure  of  mechanical  work. 

It  may  not  be  without  profit  to  consider  for  a  moment  how  it  can  be  possible 
that,  at  the  temperature  of  absolute  zero,  a  gas  can  be  compressed  without 
requiring  work  to  compress  it.  ' 

Let  us  consider  a  system  of  molecules  which  are  in  absolute  rest,  and  so  far 
distant  from  each  other  that  their  mutual  actions  can  be  disregarded.  If,  now, 
this  system  is  compressed  by  means  of  a  piston,  the  piston  will  impart  a  cer- 
tain velocity  to  the  molecules  which  it  meets  ;  but,  since  by  hypothesis  the  tem- 
perature is  kept  at  absolute  zero,  these  velocities  remain  infiniteiy  small.  We 
require  upon  the  piston,  therefore,  only  a  force  which  imparts  to  a  finite  num- 
ber of  molecules  in  a  finite  time  an  infinitely  small  velocity,  that  is,  an  infinitely 
small  force. 


*  This  is  the  law  of  "adiabatic"  expansion  or  "Poisson's  law."  See  Poisson,  Traite  de  Me- 
canique,  vol.  v.,  chap.  4 ;  Weisbach,  vol.  ii.,  art.  37ti ;  also,  page  163,  chap,  v.,  of  the  present 
volume. 


76  NOTES  AND  ADDITIONS 


NOTE  18.— (Page  37.) 

THE  NECESSARY  TENDENCY   OF   HEAT   TO   PASS   FEOM   A   WARMER  TO 
A    COLDER   BODY. 

la  a  system  composed  only  of  a  perfect  and  simple  gas,  the  tendency  of 
heat  to  pass  from  a  warmer  to  a  colder  body  is  a  necessary  consequence  of  the 
laws  of  impact  of  elastic  bodies.  We  have  seen,  in  a  preceding  note  (No.  10), 
that  in  such  bodies  the  temperature  is  proportional  to  the  living  force  of  the 
individual  molecules,  when  this  temperature  is  reckoned  from  the  absolute 
zero,  that  is,  from  —  273"  C.  It  is  at  once  evident,  that  when  different  perfect 
gases  unite,  those  molecules  which  possess  the  greatest  living  force  will  give 
up  by  impact  a  part  of  their  living  force  to  those  which  have  less  ;  or,  in  other 
word's,  heat  is  al  ways  and  of  necessity  imparted  by  the  molecules  of  the  warmer 
gas  to  those  of  the  colder. 

When,  therefore,  we  say  that  in  such  systems,  when  subjected  to  any  cycle 
of  changes  in  which  the  final  and  initial  conditions  are  the  same,  heat  can  in 
no  case  pass  from  a  cold  to  a  warmer  body,  we  simply  express  a  truth  as  clearly 
proven  as  the  impossibility  of  perpetual  motion. 

This  hardly  holds  for  other  cases.  But  we  may,  however,  presume  that 
the  general  laws  of  heat,  equilibrium,  and  motion,  are  nothing  more  than  pure 
mechanical  principles,  and  although  we  choose  a  gas  as  the  subject  of  com- 
parison, we  may  still  thus  form  some  idea  as  to  what  properly  constitutes 
equilibrium  and  difference  of  temperature  for  other  bodies.  If  a  solid  or  liquid 
body  is  of  equal  temperature  with  a  gas,  the  molecules  must  have  such  a  state 
of  motion  that,  so  long  as  the  center  of  gravity  of  the  body  is  not  changed,  the 
gas  molecules  which  come  in  contact  with  the  body  neither  receive  nor  part 
with  living  force.  It  follows,  then,  that  if  two  solid  bodies  are  in  temperature 
equilibrium  with  the  same  gas,  their  temperature  will  not  alter,  i.  e.,  the 
motions  of  their  molecules  will  not  change,  when  they  are  brought  into  direct 
contact  with  each  other.  That  which  holds  for  this  case,  holds  good  also 
when  the  ether,  upon  which  all  radiation  phenomena  depend,  is  the  medium  of 
communication. 


NOTE   19.— (Page  40.) 

THE   II^FLUEN^CE   OF    FRICTIOK   IK   THE    ELECTRO-THERMAL   USTVESTI- 
GATION^S    OF   FAVRE. 

It  is  not  necessary,  in  these  experiments,  to  take  account  of  the  influence  of 
friction,  but  we  compare  directly  the  observed  diminution  of  heat  with  the 
useful  work  of  the  engine.  The  friction  of  the  engine  undoubtedly  develops 
heat,  and  this  heat  acts  in  the  calorimeter  just  as  well  as  that  developed  by 
the  passage  of  the  current.  But  the  development  of  heat  by  friction  means  in 
reality  the  performance  of  work  and  generation  of  living  force,  and  this  double 
production  produces  an  equivalent  diminution  of  the  heat  developed  in  the 
circuit.  Thus  the  friction  increases,  on  the  one  hand,  the  heat  in  the  calori- 
meter, and  on  the  other,  diminishes  it  by  a  precisely  equal  amount.  We  need 
not,  therefore,  consider  it  at  all.  The  only  correction  is  due  to  the  friction  of 
the  rollers,  outside  of  the  calorimeter,  by  aid  of  which  a  weight  is  raised. 
Experiment  has  confirmed  the  strict  compensation  of  the  two  opposite  effects 
of  friction.  Whether  the  engine  is  at  rest,  or  whether  in  action,  without 
raising  the  weight,  we  have  always  in  the  calorimeter  the  same  amount  of 
heat. 


TO  THE  LECTURES.  77 

NOTE   20.— (Page  42.) 

THE    DISCOVERT    OF   INDUCTION    PHEN03IENA. 

The  experimeut  referred  to  is  given  in  the  Annales  de  Cliimie  et  de  Phy- 
sique, 2  Serie,  vol.  sxi.,p.  47.  An  annular  plate  of  copper  was  liung  by  a  silk 
thread  in  the  plane  of  a  circular  frame,  upon  which  was  wound  a  number  of 
turns  of  insulated  copper  wire.  A  powerful  iron  magnet  was  placed  with  one 
pole  in  the  circle  and  the  other  without. 

As  soon  as  the  current  passed,  the  circle  was  attracted  or  repelled  by  the 
electro-magnet  ;  but  the  duration  of  the  action,  as  in  all  similar  induction 
phenomena,  was  very  short.  This  fact,  probably,  prevented  Ampere  from 
feeling  confidence  in  his  experiment,  for  he  failed  to  draw  from  it  the  least 
conclusion,  and  nothing  furtlier  is  said  of  it  till  Faraday  published  his  dis 
covery.  This  is  the  more  surprising  as  Ampere,  at  the  time  when,  in  associa- 
tion with  de  la  Eive,  he  undertook  this  expeinment  (1822,  at  Genf),  sought,  in  so 
many  words  at  least,  "  to  produce  an  electric  current  by  the  action  of  another 
current."     These  are  his  words  ten  years  later. 


NOTE   21.— (Page  42.) 

DEDUCTION   OF   THE   LAWS   OF   INDUCTION    FROM   THEORY. 

We  consider  a  battery  consisting  of  any  number  of  equal  or  unequal  ele- 
ments. According  to  the  laws  of  Faraday,  the  amounts  of  chemical  action 
which  are  developed  in  the  same  time  in  tlie  difEerent  elements  are  mutually 
equivalent. 

If,  therefore,  X',  L" ,  L'",  etc.,  are  the  works  of  the  chemical  forces  in  the 
different  elements  during  the  time  required  in  each  for  the  decomposition  of 
one  equivalent  of  metal,  the  total  amount  of  heat  developed  in  the  battery  and 
conductors,  assumed  at  rest,  is 

L'  +  L'  +  L'"  H 


2X 
J  ' 

On  the  other  hand.  Joule's  experiments  have  shown  that  the  amount  of 
heat  developed  in  a  unit  of  time  in  a  conductor,  is  proportional  to  the  resistance 
and  to  the  square  of  the  intensity  of  the  current.  Let  It  be  the  total  resistance 
of  the  conducting  wire  and  battery,  y  the  intensity  of  current,  then  the  heat 
developed  in  a  unit  of  time  in  the  battery  and  conductors  is  proportional  to 
T'^B  or  to  rSA.  if  2 A  denotes  the  sum  of  the  electro-motive  forces,  and, 
according  to  Ohm's  law 

SA 
R   ' 

If  0  is  the  time  necessary  to  decompose  one  equivalent  of  metal  in  each 
element,  the  amount  of  heat  which  we  have  just  represented  by  .2i  will  be 
proportional  to  J^)  2  A,  or,  simply,  to  'S  A,  wlien  we  take  as  unit  that  intensity 
of  current  which  corresponds  to  the  decomposition  of  one  equivalent  of  metal 


78  NOTES  AND  ADDITIONS 

in  tlie  unit  of  time.  We  shall  therefore  have,  when  we  have  properly  chosen 
the  unit  of  electro-motive  force, 

Let  us  now  assume  that  the  circuit  as  a  whole,  or  that  portions  of  it,  under 
the  influence  of  outer  points  of  magnetic  attraction,  or  of  the  mutual  action  of 
the  various  elements,  moves.  Then  the  work  of  the  chemical  forces  is  equiv- 
alent to  the  heat  developed,  and  to  the  work  of  the  electro-magnetic  or  electro- 
dynamic  forces.  We  denote  by  Udt  that  part  of  this  work  which  is  performed 
in  the  indefinitely  small  time  dt. 

Let,  further,  'i  be  the  corresponding  intensity  of  the  current,  expressed  in 
the  assumed  unit.  Then  idt  is  the  fraction  of  one  equivalent  of  metal  de- 
composed in  each  element  in  the  time  dt.  Finally,  let  Qdt  be  the  total  amount 
of  heat  developed.     Then,  according  to  what  has  been  said, 

idt  -j-  =  Qdt  +  -J-  . 

Combining  Ohm's  law  with  Joule's,  we  see  that  Qdt  is  always  proportional 
to  the  product  of  idt  by  the  sum  of  the  electro-motive  forces.  It  is  impossible 
that  this  sum  should  remain  2 A.  It  is  necessary  that  by  the  action  of  motion 
it  shall  become  less.  In  other  words,  to  the  living  forces  whose  sum  is  repre- 
sented by  2 A,  we  have  an  opposing  force  F,  which  must  satisfy  the  condition 

SL  IT 

i~  =  i(:SA-F)  +  ~. 

We  shall  now  investigate  separately  the  two  cases  which  we  have  distin- 
guished. 

As  soon  as  the  circuit  (battery  included)  moves  as  a  whole,  and  without 
changing  its  shape,  under  the  influence  of  outer  force  centers,  the  elementary 
work  Udt  is  proportional  to  the  energy  C  of  these  force  centers,  to  the  intensity 
i  of  the  stream,  and  to  a  function,  (p,  which  depends  upon  the  relative  position 
of  the  circuit  and  these  force  centers  at  any  given  moment,  upon  the  kind  of 
motion,  and  upon  the  distance  vdt  passed  through  by  any  element.  We  have, 
therefore, 

^  =  2A-F+^, 

■n  GcpV 

The  factor  v  is  the  velocity  at  a  given  moment.  We  see  therefore  that  the 
electro-moti  ve  force  of  induction  is  proportional  to  the  velocity  of  displacement, 
and  to  the  expression  Cq>,  which,  when  multiplied  by  vdt,  gives  the  elementary 
work  of  the  outer  forces  upon  a  circuit  traversed  by  a  current  whose  intensity 
is  unity. 

When  the  elements  of  the  circuit  change,  by  reason  of  their  mutual  action, 
we  can  represent  the  elementary  work  of  their  mutual  actions  by  i'-tpvdt,  where 
^  is  a  similar  function  to  (p.     Therefore, 

The  electro-motive  force  in  this  case  is  proportional  to  the  intensity  of  the 
stream  and  to  the  velocity  of  the  relative  motion. 

In  the  general  case  of  change  of  form,  and  total  or  partial  displacement  by 
th€»influence  of  outer  forces,  the  electro-motive  force  of  induction  is  the  sum 
of  the  two  preceding  expressions. 


TO  THE  LECTURES.  79 


NOTE  22.— (Page  42.) 

THE   COMPLETE   TEAE"SFOEMATIO]Sr     OF     HEAT    INTO    WORK,    BY    THE 
ELECTRO-MAGNETIC   ENGINE. 

Let  us  consider  an  electro-magnetic  engine.  We  assume  first  that  only  the 
immovable  pieces  are  traversed  by  the  current,  and  that  the  movable  pieces  are 
permanent  magnets.*  We  assume  such  an  engine,  which,  under  tiie  influence 
of  outer  resistance,  has  attained  its  condition  of  normal  activity,  so  that  in 
successive  periods  the  rotation  is  identical.  This  condition  does  not  include, 
strictly  speaking,  uniformity  of  motion  ;  but,  in  a  well -constructed  machine,  in 
which  the  intensity  of  the  mutual  action  of  the  magnets  and  wire  spirals  varies 
from  one  moment  of  the  rotation  to  another  but  little,  the  rotation  can  be  con- 
sidered as  essentially  uniform.  If  we  call  Fthe  velocity  of  this  rotation,  the 
electro-motive  force  of  induction  is^F,  where  ^  is  a  constant  coefficient  which 
depends  upon  the  strength  of  the  moving  magnets  and  the  arrangement  of  the 
machine.  Hence,  if  we  denote  by  A  the  sum  of  the  electro-motive  forces,  by 
R  the  resistance,  and  by  i  the  intensity,  we  have 

.       A-KY 


The  heat  corresponding  to  the  decomposition  of  one  equivalent  of  metal  in  each 
element  is,  therefore, 

A  -KY. 

In  rest,  it  would  be  A.     The  heat  transformed  into  work  is  then  KY.     The 

KY 

ratio  —J-  of  these  two  quantities  increases  with  the  velocity,  and  approaches 

unity  as  the  electro-motive  force  A  —  ^Fand  the  intensity  approaches  zero. 
If  the  movable  and  immovable  parts  are  traversed  by  the  same  current,  the 
electro-motive  force  of  induction  is  expressed  by  h  Yi,  so  that 

.      A-hYi  .  A 


R         '  '  R  +  hY' 

The  amount  of  heat  developed  per  unit  of  time  in  the  circuit  is,  therefore, 

(  -^ r^ir  )   -">  or,  I A  — =-r=;  . 

\R  +  7iY/  R  +  hY 

During  the  time  0,  which  is  required  for  the  decomposition  of  one  equivalent 
of  metal  in  each  element,  the  heat  developed  is 

7?  P 

iOA 


R  +  hY'  '  R  +  hY' 

since  we  assume  (see  preceding  note)  that  iO  is  equal  to  unity.     In  a  state  of 
rest,  this  quantity  is  A.     The  heat  transformed  into  work  is  then 

R  +  hY' 
which  approaches  A  the  greater  F  becomes. 

*  Fromment  has  often  constnicfed  machines  of  this  kind.  Tiie  theorj'  of  those  in  which  the 
immoA-able  pieces  are  magnets  and  the  movable  ones  wire  coils,  does  not  differ  essentially  from 
the  present  presentation. 


80  NOTES  AND  ADDITIONS 


NOTE  23.— (Page  43.) 

DETERMINATION"    OP    THE    MECHANICAL    EQUIVALENT   OF    HEAT    BY 
ELECTRO-MAGNETS. — (  JOULE. ) 

Joule  caused,  by  means  of  a  weiirlit,  a  movable  electro-magnet  to  turn 
between  the  poles  of  an  immovable  electro-magnet  of  great  power.  He  deter- 
mined first  the  weight  which  was  necessary  in  order  to  give  to  the  apparatus 
a  constant  velocity  under  the  influence  of  friction,  the  current  of  both  electro- 
magnets being  open.  Then  the  conducting  wire  of  the  fixed  electro-magnet 
was  connected  with  the  battery,  and  that  of  the  movable  closed  with  a  short 
thick  wire,  and  the  weight  determined  which  had  to  be  added  to  maintain  the 
same  constant  velocity,  as  also  the  heat  developed  in  the  movable  current. 

This  last  part  of  the  experiment  appears  to  have  left  much  to  be  desired. 
The  movable  electro-magnet  was  placed  in  a  glass  vessel  filled  with  water,  and 
the  rise  of  temperature  of  this  compound  system  directly  observed,  in  order  to 
find  the  heat  generated.  Two  constant  sources  of  error  must  tend  to  cause 
this  determination  to  give  too  small  values.  First,  it  is  extremely  doubtful 
whether  there  is  simultaneously  a  common  temperature  in  the  water  and  the 
soft  iron  and  the  insulated  copper  wire  which  form  the  movable  system.  More- 
over, the  long  cylindrical  shape  of  the  system  favors  the  cooling  by  radiation 
and  contact  with  the  air.  This  last  is  also  increased  by  the  rotary  motion. 
Whatever  care  is  taken  in  applying  corrections,  it  can  hardly  be  avoided 
estimating  too  low  the  heat  developed  by  the  given  expenditure  of  work,  and 
hence  obtaining  too  large  a  value  for  the  mechanical  equivalent.  It  is,  there- 
fore, not  surprising,  that  the  value  deduced  from  these  experiments  is  about 
-jVth  greater  than  the  probable  value.  In  some  special  experiments,  the  differ- 
ence is  even  still  greater. 


NOTE  24.— (Page  45.) 

THE     NATURE     OF   ELECTRO  -  MAGNETIC     AND     ELECTRO  -  DYNAMIC 

FORCES. 

It  may  be  objected  that  we  have  made  use  in  our  lectures  of  the  principle 
of  the  impossibility  of  perpetual  motion  as  an  absolute  truth.  It  may  be  said 
that  we  apparently  forget  that  there  are  natural  forces,  such  as  electro-mag- 
netic and  electro-dynamic,  which  do  not  depend  alone  upon  mass  and  distance  ; 
that  there  are  forces  with  whose  help  therefore  we  can  cause  in  certain  cases 
rotary  motion,  the  velocity  of  which  may  be  indefinitely  accelerated.  We  had 
the  intention  of  noticing  this  objection  in  the  second  lecture,  when  speaking  of 
the  electro-magnetic  engine  ;  but  it  appeared  better,  on  consideration,  to  reserve 
it  for  a  note. 

Let  us  consider  first  the  electro-magnetic  forces.  Experiment  shows  that 
magnets  act  upon  currents,  and  inversely.  All  the  effects  of  these  actions  can 
be  referred  to  a  system  of  forces  which  afiect  the  different  elements  of  the 
current,  which  depend  not  only  upon  the  distances,  but  also  upon  certain  angles, 
and  which  do  not  act  in  the  straight  lines  connecting  the  current  elements  and 
the  magnetic  centers. 

For  a  closed  circuit  of  invariable  form,  this  system  of  forces  can  be  replaced 
by  an  equivalent,  which  is  apparently  entirely  different  from  the  preceding, 
and  consists  of  forces  which  satisfy  the  ordinary  conditions  of  action  of  natural 
forces.  In  this  case,  the  difficulty  disappears  at  once  of  itself.  This  substitu- 
tion is  however  no  longer  possible  when  the  circuit  is  not  closed,  or  when,  in 
other  words,  the  closed  conductors  traversed  by  the  current  consist  of  several 


TO   TEE  LECTURES.  81 

independent  portions.  Tlie  motion  of  each  of  these  parts  depends  solely  upon 
forces  which  act  upon  its  various  elements,  and  it  is  clear  that  this  motion,  under 
certain  circumstances,  is  one  of  rotation,  which,  without  the  influence  of  friction, 
resistance  of  the  air,  and  similar  resistances,  would  be  infinitely  accelerated. 

Ampere  has  repeatedly  declared  that  here  is  an  actual  exception  to  the 
general  laws  of  mechanics.  There  is  no  treatise  or  presentation  of  any  com- 
pleteness, upon  electro  magnetism,  in  which  this  is  not  illustrated.  There  is 
even  no  elementary  presentation  in  which  the  fact  of  indefinitely  accelerated 
rotation  is  not  in  various  ways  experimentally  shown.  But  error  is  committed, 
and  any  presentation  must  be  very  imperfect,  if  anything  real  is  seen  in  this 
apparent  exception.  We  will  consider  one  of  the  simplest  experiments  of  this 
character,  which  is  to  be  found  in  all  regular  courses  in  Physics.  A  small 
rectilinear  horizontal  current  turns  about  a  vertical  axis  through  one  of  its 
ends,  under  the  action  of  a  vertical  magnet,  situated  in  the  prolongation  of  the 
axis. 

It  requires  but  little  attention  to  recognize  that,  at  the  end  of  each  revolu- 
tion, the  velocity  is  somewhat  greater  than  at  first,  at  least  as  long  as,  under 
the  influence  of  the  resistances,  the  maximum  is  not  attained.  Perpetual 
motion  seems  therefore  attained,  since,  at  the  end  and  at  the  beginning  of  each 
revolution,  the  position  of  the  current  and  the  magnet  is  the  same.  But  does 
this  coincidence  of  position  include  the  condition  that  nothing  is  changed  in 
the  total  system  of  mutually  interacting  bodies  1  This  system  consists  not  only 
of  the  current  and  the  movable  magnet,  but  also  of  the  battery  which  sets  the 
electric  current  in  motion,  and  the  conductors  which  unite  the  battery  with  the 
two  ends  of  the  movable  current.  We  will  not  speak  of  the  special  phenomena 
which  occur  at  the  point  of  contact  of  movable  and  immovable  parts.  The 
battery  is  the  seat  of  incessant  transformation  of  chemical  actions.  Is  it  there- 
fore strange  that  such  transformation  should  cause  a  continuous  increase  of  the 
velocity  of  rotation  of  a  movable  wire  ?  The  actual  mechanism  by  which  the 
phenomena  are  caused  is  unknown,  but  nothing  necessitates  us  to  admit  that 
the  action  of  the  real  elementary  forces  does  not  follow  the  general  laws  of 
action  of  natural  forces.  The  assumed  elementary  forces,  to  which  we  are 
necessarily  led  when  we  limit  our  consideration  to  the  magnet  and  movable 
current,  are  functions  of  the  angle,  and  perpendicular  to  the  plane  of  the  mag- 
net and  current.  These  forces  are  not  in  the  least  analogous  to  the  elementary 
forces  which  govern  the  motions  of  the  stars  'or  the  fall  of  bodies.  They  are 
pure  mathematical  symbols,  which  represent  not  the  reality,  but  only  the  last 
stages  to  which,  thus  far,  analysis  of  the  phenomena  has  led  us.  We  can  say 
the  same  of  the  electro-dynamic  forces,  and  of  the  famous  formula  by  which 
Ampere  has  represented  what  he  calls  the  opposite  action  of  two  current  ele- 
ments. This  formula  is  an  experimental  law,  which,  in  its  unlimited  fruitf ul- 
ness,  indeed,  exhausts  every  possible  variation  of  the  phenomena ;  but  which 
possesses  no  reality  outside  of  the  circle  of  phenomena  for  which  it  forms  the 
general  bond.  If,  for  example,  it  were  possible  to  place  two  current  elements, 
independently  of  any  voltaic  circuit,  in  the  same  physical  condition  as  when 
they  form  an  actual  part  of  such  a  circuit,  it  proves  nothing,  that,  in  accord- 
ance with  Ampere's  laws,  they  miust  approach  or  recede  fiom  each  other.  All 
that  we  can  assert  is  that  these  laws  represent  the  phenomena  in  all  cases  open 
to  experiment.  We  can  see  in  them  only  the  translation  of  the  secret  mechan- 
ism by  which  the  phenomena  are  produced,  and  nothing  prevents  the  admis- 
sion that  the  actual  forces  involved  in  such  mechanism  are  simple  functions 
of  the  distances,  and  act  in  the  line  connecting  any  two  mutually  acting  points. 

This  was,  moreover.  Ampere's  own  view  of  his  discoveries.  If  he  seldom 
referred  to  it,  even  sometimes  apparently  rejected  it  for  the  opposite,  it  was 
only  not  to  offend  the  scientific  views  of  his  contemporaries,  who  without  it  had 
difficulty  enough  to  appreciate  his  experiments,  and  who  would  have  rejected 
his  hypotheses  without  proof.  But,  in  the  remarks  which  he  has  added  to  the 
presentation  of  his  theory  (read  at  the  official  session  of  the  Academy,  April  8, 
1822),  he  has  expressed  himself  in  a  manner  which  allows  no  doubt  as  to  his 
convictions. 

"  I  remarked,"  he  says,  "1.  That  the  attractions  and  repulsions,  whose 
6 


82  NOTES  AND  ADDITIONS 

existence  between  portions  of  the  conducting  wires  I  had  recognized,  could  not 
arise  in  the  same  way  as  that  of  ordinary  electricity,  by  reason  of  unequal  dis- 
tribution of  the  two  fluids  which  mutually  attract  each  other,  and  every  part  of 
which,  of  the  same  kind,  is  repelled,  since  all  the  hitherto  known  properties  of 
the  conducting  wires  show  that  neither  tbe  one  nor  the  other  of  the  two  fluids 
occurs  in  larger  quantities,  in  a  body  which  serves  as  conductor  of  an  el9ctrical 
current,  than  when  the  same  bodies  are  in  their  natural  condition.* 

"  2.  I  remarked  that  it  is  difficult  not  to  conclude,  therefore,  that  these  attrac- 
tions and  repulsions  are  caused  by  the  extremely  rapid  motion  of  the  two  elec- 
tric fluids  which  traverse  the  conductor,  by  reason  of  almost  instantaneous  de- 
composition and  composition  in  opposite  directions  ;  a  motion  assumed  by  all 
physicists  since  Volta,  and  which  the  theory  given  by  this  renowned  savant  of 
the  admirable  instruments  constructed  by  him,  substantiates. 

"3.  If  we  ascribe  the  attractions  and  repulsions  of  the  conducting  wires  to 
this  cause,  we  cannot  avoid,  if  we  explain  the  ordinary  electrical  phenomena  in 
the  customary  manner,  admitting,  further,  that  the  motions  of  the  two  electrici- 
ties in  the  wires  are  propagated  in  every  direction  in  the  neutral  fluid  formed 
by  this  union,  with  which,  necessarily,  all  sj^ace  must  be  filled  ;  so  that  when 
the  motions  thus  arising  in  the  surrounding  medium,  caused  by  two  small 
current  portions,  mutually  coincide,  there  is  a  tendency  to  approach,  which  is, 
in  fact,  the  case  when  we  observe  attraction  ;  and  that  when  the  two  motions  are 
opposed,  the  two  current  portions  tend  to  repel  each  other,  as  experiment  also 
shows. 

"  4.  If  we  consider  these  attractions  and  repulsions  as  actually  caused  by  these 
reasons,  the  law  that  a  small  portion  of  the  electric  current  can  be  replaced  by 
two  others,  which  stand  to  it  in  the  same  relation  as  two  forces  to  their  result- 
ant, is  a  necessary  consequence  of  this  assumption  ;  since  velocities  are  com- 
posed like  forces,  and  since  the  motion  which  the  small  portion  of  a  current, 
represented  in  intensity  and  direction  by  the  resultant,  imparts  to  the  fluid 
which  fills  space,  is  necessarily  equal  to  that  which  is  caused  in  the  same  fluid 
by  the  union  of  the  two  small  current  portions  which,  in  similar  manner,  are 
represented  by  the  two  components. 

"At  the  time  when  I  was  occupied  by  these  ideas,  Fresnel  communicated 
to  me  his  elegant  researches  upon  light,  from  which  he  deduced  the  laws  which 
determine  all  the  conditions  of  optical  phenomena. 

"  I  was  surprised  at  the  agreement  between  his  views  and  those  to  which 
I  had  been  led  by  the  consideration  of  electro  dynamic  attractions  and  repul- 
sions. 

"  He  showed,  from  the  accordance  of  these  phenomena,  that  the  ethereal 
fluid  of  space,  which  cannot  be  regarded  as  the  result  of  the  union  of  the  two 
electricities,  must  be  nearly  incompressible,  and  must  permeate  all  bodies,  as 
gas  flows  through  a  net,  and  that  the  motions  caused  in  this  fluid  must  be  prop- 
agated by  a  kind  of  friction,  which  enables  the  moving  layers  to  set  in  motion 
others.  Hence  it  was  natural  to  suppose  that  the  flowing  electric  current  in  a 
conducting  wire  caused  the  surrounding  neutral  fluid  to  lake  part  in  its  motion, 
and  in  part  rubbed  on  it,  so  that  a  reaction  of  this  fluid  on  the  current  was 
caused.  This  reaction  can  cause  no  tendency  to  a  displacement  of  the  wire,  so 
long  as  the  difference  of  velocities  on  all  sides  of  the  wire  is  the  same.  There 
will  be,  however,  a  tendency  to  move  the  wire  as  soon  as  a  second  current  ex- 
ists, and,  indeed,  either  toward  the  side  on  which  this  difference  of  velocity, 
and  hence  of  reaction,  is  less— that  is,  the  side  upon  which  another  electric  cur- 
rent tends  to  move  the  fluid  in  the  same  direction — or  toward  the  opposite  side, 
upon  which  this  difference  is  greater,  because  there  another  electric  current 
exists  which  tends  to  move  the  fluid  in  the  opposite  direction,  according  as  the 
two  mutually  acting  currents  flow  in  the  same,  or  in  opposite  directions. 

"  These  views  certainly  make  clear  the  attraction  between  similarly  flowing, 

*  We  know  now  that  free  electricity  exists  upon  the  surface  of  conductors  through  which  a 
current  flows  The  distribution  of  this  electricity  is,  however,  such  that  it  has  no  influence  upon 
tlie  electro-dynamic  phenomena.  Moreover,  by  the  composition  of  forces,  which  are  only  func- 
tions of  tlie  distances,  we  could  never  obtain  resulrants  which  are  functions  of  the  angles. 


TO   THE  LECTURES.  83 

and  the  repulsion  between  oppositely  flowing  currents,  in  accordance  with  ex- 
periment ;  but  I  have  not  forgotten  the  fact^hat,  as  it  is  not  possible  to  calcu- 
late all  the  effects  of  the  motion  of  fluids,  they  are  too  general  to  serve  as  the 
foundation  for  a  law  whose  correctness  can  be  conlirmed  by  direct  and  exact 
experiment.  This  is  the  reason  why  I  have  confined  myself  to  representing  it 
simply  as  a  fact  based  upon  observation." 

Thus  far  Ampere.  It  is  interesting  to  see  how  the  renowned  author  of  the 
"  Theorie  des  phenomenes  electrodynamiques  "  recognized  back  of  the  problem 
solved  by  him,  another  still  deeper  and  more  ditBcult,  the  solution  of  which  he 
left  to  the  future.* 


NOTE  25.— (Page  46.) 

ELECTEOLYTIC    CONVECTIOlSr. 

When  a  current  is  made  to  perform  decomposition  of  water,  the  chemical 
heat-equivalent  of  the  decomposition  must  be  abstracted  from  the  heat-equiv- 
alent of  the  electrolytic  process  in  the  galvanic  battery.  Since,  now,  for  the 
decomposition  of  one  equivalent  of  metal  in  each  element  of  the  battery,  one 
equivalent  of  water  is  decomposed  in  the  voltameter,  a  decomposition  of  water 
can  only  occur  when  the  electrolytic  processes  in  the  battery  develop  more  heat 
than  can  be  generated  by  the  reunion  of  the  oxyhydrogen  gas  developed  in  the 
voltameter.  About  If  Daniell 's.  cells  are  necessary  to  give  continuous  decom- 
position of  water,  f 

With  a  single  Daniell  cell,  therefore,  no  decomposition  of  water  is  possible. 

The  conditions  are  essentially  different  when  both  electrodes  and  the  liquid 
of  the  voltameter  are  completely  saturated  with  hydrogen  or  with  oxygen. 

There  is  then  a  transmission  of  electricity  from  one  electrode  to  another, 
either  through  a  non-electrolytic  conduction  of  the  water,  or  through  a  process 
which  Helmhollz  calls  electrolytic  convection.  This  peculiar  process  consists  in 
the  fact  that,  under  such  conditions,  an  electrolytic  decomposition  of  the  jvater 
and  a  separation  of  hydrogen  and  oxygen  can  take  place.  If  thus,  for  example, 
the  voltameter  is  completely  saturated  with  hydrogen,  the  oxygen  combines  im- 
mediately, upon  its  generation,  with  the  condensed  hydrogen  upon  the  surface 
of  the  lalatinum.  Then  the  negative  work  of  the  water  decomposition  is  com- 
pensated by  the  positive  work  of  the  water  formation  on  the  one  electrode.  In 
this  case  the  water  decomposition  is  connected  with  no  essential  consumption 
of  heat.  The  condition  of  such  a  process  is,  therefore,  that  on  one  electrode 
more,  and  upon  the  other  correspondingly  less,  hydrogen  occurs.  The  entire 
process  is  thus  limited  to  a  different  distribution  of  the  gas  contained  in  the 
liquid. 

Thus  Helmholtz  found  that  a  current,  which  was  able  to  decompose  in  24 
hours  60  milligrams  of  silver,  could  pass  for  a  day,  without  diminution  of  its 
strength,  through  such  a  voltameter  saturated  with  hydrogen,  without  causing 
more  than  a  just  appreciable  polarization. 

Especially  under  very  low  pressure,  for  more  rapid  development  of  hydrogen, 
the  hydrogen  separated  in  gaseous  state. 

With  such  a  voltameter,  the  pair  of  platinum  plates  being  laden  Avith  hydro- 
gen gas,  a  development  of  hydrogen  may  be  caused  by  one  Daniell's  cell. 

This  phenomenon  was  earlier  noticed  by  Poggendorff.  It  finds  its  explana- 
tion in  Helmholtz's  experiments,  and  does  not  stand,  as  we  see,  in  contradiction 
to  the  principle  of  equivalence. 

*  The  complete  solution  of  the  problem  has  been  essayed,  in  recent  times,  by  Helmholtz  in  his 
paper  "UeberdieBewesinnCTsgleichnngender  Electricitat-t  fur  ruhende,  leitende  Korper,"  Crell's 
Journal,  vol.  Ixxi.,  p.  57,  and  by  Carl  Neumann,  in  his  work  "  Theorie  der  elelitrischen  Krafte," 
1873. 

t  Thomson— On  the  Mechanical  Theory  of  Electrolysis,  Phil.  Mag.,  1851.  He  gives  the  quan- 
tity at  1.318  Daniell  cells. 


84  NOTES  AND  ADDITIONS 

NOTE  26.— (Page  46.) 

UPON   THE   POLARIZATION   OF   THE   ELECTEODES. 

We  may,  by  the  aid  of  the  same  mechanical  consideration,  deduce  the  neces- 
sity of  another  phenomenon,  viz.,  that  of  the  polarization  of  the  electrodes. 

When  the  circuit  is  completely  metallic  and  remains  immovable,  the  heat 
developed  in  a  given  time  represents  the  total  work  of  the  chemical  forces. 
When  the  circuit  also  contains  a  compound  liquid,  the  heat  developed  in  the 
cells  by  the  same  amount  of  chemical  action  must  be  less,  since  it  icpresents 
only  the  excess  of  the  positive  work  in  the  voltaic  cells  over  the  negative  work 
in  the  decomposing  apparatus.  It  is,  therefore,  necessary  that  this  heat  shall 
be  less  than  that  which  is  obtained  when  the  liquid  is  replaced  by  a  metallic 
conductor  of  the  same  resistance.  This  can,  however,  only  be  the  case  when 
the  liquid  changes  the  current  intensity  in  some  other  manner  than  by  the  intro- 
duction of  its  resistance. 

Since,  now,  we  know  that  there  are  no  means  of  diminishing  the  inten- 
sity of  a  current  other  than  increasing  the  resistance  of  the  conductor,  or  dimin- 
ishing the  electro-motive  force,  we  see  that  the  introduction  of  such  a  liquid 
must  have,  as  an  immediate  and  necessary  consequence,  a  diminution  of  the 
total  electro-motive  force  ;  that  is,  the  development  of  an  electro-motive 
counter-force. 

Upon  this  rests  directly  the  polarization  of  the  electrodes.  In  consequence 
of  this  polarization,  the  current  of  a  single  cell  of  the  ordinary  battery  reduces, 
by  the  introduction  of  a  voltameter  with  acid  water,  to  zero,  and  hence  the 
decomposition  of  water  under  these  circumstances  is  impossible.  When  the 
liquid,  during  its  decomposition  by  the  action  of  one  of  the  chemical  elements 
originated  by  the  decomposition,  is  again  formed  upon  the  corresponding  elec- 
trode, the  work  of  the  chemical  forces  is  actually  zero,  and  we  know  that  then 
no  polarization  can  occur. 


KOTE   37. —(Page  46.) 

THE   DECOMPOSITION  OF   ZINC   IN   DILUTE  ACIDS. 

It  has  been  long  observed  that  when  commercial  zinc  is  dissolved  in  acid 
water,  the  generation  of  hydrogen  does  not  take  place  at  all  points  of  the 
metal,  but  at  certain  special  points,  which  appear  thus  to  be  different  from  the 
others. 

De  la  Rive  has  observed  that  these  points  are  fewer  with  distilled  zinc,  and 
that  the  development  of  hydrogen  takes  place  more  slowly  than  for  ordinary 
zinc.  Finally,  Almeida  has  found,  after  he  had  succeeded  in  producing  per- 
fectly pure  zinc  by  galvanic  process,  that  this  metal  resisted  perfectly  the 
action  of  dilute  sulphuric  acid.  In  both  cases  the  pure  zinc  assumed  the  prop- 
erties of  the  ordinary  metal  when  some  other  metal  was  added,  so  that  the 
acid  came  in  contact  with  a  surface  not  homogeneous  in  character. 


NOTE   28.— (Page  47.) 

UPON   THE   APPLICATION  OP  THE  MEASUREMENT  OF  ELECTRO-MOTIYE 
FORCES   TO   THERMO-CHEMICAL   INVESTIGATIONS. 

In  the  first  part  of  Note  32  we  have  said  that  the  heat  developed  in  a  given 
time  by  a  current  in  its  total  circuit,  is  proportional  to  the  product  of  the  inten- 
sity and  the  sum  of  the  electro-motive  forces.     If  we  consider  different  cir- 


TO   THE  LECTVRE8.  85 

cuits,  each  of  whicli  consists  only  of  a  single  pair  and  metallic  conductors,  tlic 
amounts  of  heat  developed  in  a  unit  of  time  in  these  different  circuits  are  to 
each  other  as  the  products  of  the  intensity  and  electro-motive  forces  of  each 
pair.  Since,  however,  the  intensity  is  proportional  to  the  number  (whole  or 
fractional)  of  metal  equivalents  decomposed,  it  follows  that  the  heat  developed 
by  the  decomposition  of  one  equivalent  of  metal  in  the  different  elements,  is 
directly  as  the  electro-motive  force  itself.  We  can,  therefore,  replace  calori- 
metric  measurements  by  measurements  of  the  electro-motive  forces,  provided 
that  we  know  in  a  few  cases,  by  direct  experiment,  for  a  certain  amount  of 
heat  developed,  the  corresponding  electro-motive  force. 

The  practical  advantage  of  this  method  is  apparent,  but  its  application 
involves  some  difficulties.  In  all  those  cases  in  which  the  chemical  action 
which  causes  the  current  is  accompanied  by  a  development  of  gas,  the  electro- 
motive force  varies  with  the  intensity  of  the  current.  But,  by  local  heat  phe- 
nomena occurring  at  those  points  at  which  the  gas  is  generated,  the  case  may 
happen  that  the  total  heat  production  is  constant.  We  cannot  therefore  speak, 
without  specifying  further,  of  any  proportionality  between  the  two  quantities. 
Many  observations  made  with  care  and  skill,  because  no  account  was  taken  of 
these  circumstances,  have  lost  the  greater  part  of  their  value. 


NOTE   29.— (Page  48.) 

THE   INFLUENCE    OF   THE    FRICTION   OF   THE   BLOOD   UPON  THE 
ANIMAL    HEAT. 

These  views  hold  good  in  spite  of  the  interior  motions  in  organisms,  and  in 
spite  of  the  resistances  which  these  encounter.  There  is  no  reason  for  taking 
account  of  that  part  of  these  resistances  due  to  the  action  of  the  outer  forces — 
— as,  for  example,  gravity — so  long  as  there  is  no  displacement  of  the  center  of 
gravity  of  a  body.  The  interior  circulation  of  fluids,  the  movements  of  the 
muscles  resulting,  the  elastic  reactions  of  the  vessels,  cannot  give  rise  to  any 
work  of  gravity. 

As  to  the  inner  resistances,  these  are  the  frictions  which  must  develop  just 
as  much  heat  as  the  mtiscular  force,  which  maintains  the  motion  of  the  liquids 
in  spite  of  friction,  consumes.  We  see  then  how  useless  is  the  investigation 
of  the  influence  of  the  friction  of  the  blood  in  the  vessels  upon  the  heat  of 
animals,  which  some  physiologists  have  made.  In  order  to  overcome  this  fric- 
tion, the  action  of  the  heart  is  necessary.  In  order  to  maintain  this  action,  a 
portion  of  the  heat  furnished  by  the  interior  combustion  in  the  organism  is 
necessary.  This  loss  of  heat  is,  however,  completely  replaced  by  the  heat  gen- 
erated by  the  friction  of  the  blood  in  the  total  circulatory  system.  There  is 
thus  only  another  distribution  of  the  heat,  while  its  total  amount  remains 
unchanged. 

So  long  as  the  animal  remains  at  rest,  we  are  perfectly  justified  in  compar- 
ing this  total  heat  quantity  with  the  sum  of  the  chemical  actions  arising  from 
respiration.* 


NOTE   30.— (Page  49.) 


UPON  VEGETATION  WHICH  IS  CAREIED  ON  WITHOUT  THE  INFLUENCE 
OF   LIGHT. 

When  the  influence  of  light  is  withdrawn  from  the  higher  plants,  two  cases 
may  occur  :  either  they  may  act  like  inanimate  bodies,  absorbing  oxygen  from 

*  See  Hirn.  Remarqnes  snr  le  role  reel  que  joiie  le  frottement  des  muscles  dans  le  plieuomere 
de  la  calorification  des  etres  vivants  a  sang  cliaud  ou  a  sang  froid.  Cosmos,  1862,  vol.  xxi., 
p.  257. 


86  NOTES  AND  ADDITIONS 

the  air  and  allowing  the  water  and  carbonic  acid  in  the  soil  to  filter  through 
their  organism,  then  bleaching,  and  if  often  increasing  in  their  dimensions, 
still  the  proportion  of  combustible  substances  seeming  rather  to  diminish  than 
to  increase  ;  or  a  part  of  their  tissues  is  destroyed  by  more  or  less  rapid  oxida- 
tion, and  experiences  far-reaching  changes,  which  nevertheless  do  not  require 
the  action  of  any  outer  forces  ;  these  can  be  regarded  as  oxidations  brought 
about  by  the  natural  activity  of  the  affinities,  as,  for  example,  in  the  gennina- 
tlon  of  seed. 

The  question  now  arises  whether  the  same  is  true  of  the  lower  plants,  whose 
life  is  almost  entirely  independent  of  the  influence  of  light,  or  if  not,  in  what 
way  is  this  influence  replaced,  and  how  is  it  possible  that  they  grow,  and  that 
their  vegetation  is  accompanied  by  a  negative  work  of  the  affinities. 

In  its  present  state,  experimental  physiology  gives  no  reliable  answer  to 
these  questions.  In  order  to  answer  them,  we  must  first  have  exact  compara- 
tive analyses  of  lower  plants  which  have  completed  their  development,  and  we 
must  investigate  chemically  the  materials  by  the  use  of  which  they  have  de- 
veloped. In  most  cases  these  materials  are  decomposing  organic  bodies,  and  it 
is  possible  that  the  simple  elements  composing  every  organism,  such  as  carbon, 
hydrogen,  oxygen,  nitrogen,  occur  in  these  in  the  same  proportions  as  in  the 
plants  themselves,  but  in  a  different  grouping. 

The  vegetable  life  can,  therefore,  only  be  a  series  of  equivalent  transforma- 
tions, which  demand  no  expenditure  of  work  furnished  by  outer  forces. 

If,  on  the  other  hand,  experiment  shows  that  in  the  tissues  of  the  lower 
plants,  without  any  action  of  light,  carbon  and  hydrogen  exist  in  relatively 
higher  proportions  than  in  the  organic  substances  upon  which  they  live,  we 
may,  it  seems  to  me,  account  for  it  somewhat  as  follows.  Almost  always  dur- 
ing the  development  of  such  plants,  the  organic  bodies  which  serve  as  nourish- 
ment are  decomposed,  and  pass  gradually  into  a  condition  in  which  they  tend 
to  follow  the  natural  activity  of  the  affinities.  There  is  thus  evidently  a  posi- 
tive work  of  the  affinities,  and  hence  a  production  of  heat.  Is  it  not  possible 
that  a  part  of  this  heat  is  made  use  of  by  the  plant  itself,  and  causes  phenom- 
ena which  correspond  to  a  negative  work  of  the  affinities?  In  this  way  the 
action  of  the  sun's  rays  may  be  replaced.  It  seems  as  if  an  observation  of 
Pasteur  gave  a  certain  probability  to  this  view.  Pasteur  has  shown  that  the 
formation  of  acid  in  alcohol  is  brought  about  by  the  oxygen,  which  the  count- 
less organisms  living  upon  the  surface  of  the  liquid  condense.  When  these 
plants  are  not  present,  the  oxygen  of  the  air  is  not  capable  of  oxidizing  the 
alcohol  ;  when  the  oxygen  is  absent,  these  plants  cannot  live. 

The  oxidation  does  not  appear  to  proceed,  however,  from  any  special  activity 
of  the  plants,  but  seems  only  to  depend  upon  their  presence  and  the  property 
which  they  possess  in  a  considerable  degree  of  condensing  gases  upon  the 
surface.  It  is  not  therefore  reasoning  in  a  circle,  when  we  assume  that  the 
oxidation  of  the  alcohol  is  a  necessary  condition  for  the  acid-forming  vegeta- 
tion. It  would  even  be  quite  natural  to  suppose  that  the  heat  developed  by 
this  oxidation,  and  which  is  so  considerable  that  no  thermometer  is  required  to 
detect  it,  is  in  part  made  use  of  for  the  production  of  such  phenomena  of  vege- 
table life,  whicli  are  opposed  to  the  tendency  of  the  affinities. 

In  the  life  processes  of  the  barm  fungus  there  seems  something  similar. 
The  sugar  is  decomposed  during  the  fermentation  into  alcohol  and  carbonic  acid. 
The  alcohol  possesses  a  considerably  less  heat  of  combustion  than  the  quantity 
of  sugar  which  is  necessary  for  its  formation.  There  is,  therefore,  work  or 
living  force  performed  in  this  decomposition.  A  part  of  this  living  force  is 
applied  in  calling  forth  the  chemical  processes  which  are  involved  i<i  the  forma- 
tion of  the  cells  of  the  fungus.  Another  part  is  directly  transformed  into  heat. 
On  the  one  side  we  have,  during  fermentation,  a  process  which  corresponds  to 
the  natural  tendency  of  the  affinities  ;  on  the  other,  the  satisfaction  of  the 
chemical  force  of  attraction  is  the  source  of  the  positive  work,  upon  which  the 
fungus  draws  in  order  to  form  new  cells,  or  perform  negative  work. 

We  see,  therefore,  that  by  fermentation,  likewise,  cells  are  formed,  plant 
organisms  increase,  and  not  inconsiderable  quantities  of  heat  are  developed 
without  any  outer  force,  such  as  light,  as  the  cause.* 

*  See  A.  Mayer,  Pogg.  Auu.,  vol.  cxlii,  p.  293. 


TO   THE  LECTURES. 


87 


NOTE  31.— (Page  49. 


THE  ABSOEPTIOK  SPECTKUM  OF  CHLOEOPHYLL,   AND  THE  INFLUENCE 
OF   COLORED    LIGHT   UPON   THE   GEOWTH   OF   PLANTS. 

As  long  as  the  action  of  the  sun's  rays  upon  plants  was  not  recognized  as 
the  cause  of  those  processes  which  continually  go  on  in  the  growth  of  vegetation, 
it  must  have  seemed  very  puzzling  from  whence  the  enormous  amounts  of  liv- 
ing force  could  originate,  which  are  accumulated  ready  for  work,  in  plants. 
Now  we  know  that  these  processes,  which  are  opposed  to  the  natural  tendency 
of  the  chemical  alfinities  (with  very  few  exceptions,  noticed  in  Note  30)  take 
place  only  under  the  action  of  light,  and,  indeed,  of  light  upon  those  parts  of 
plants  which  contain  chlorophyll. 

The  green  chlorophyll-containing  plants  take  from  the  air  the  carbon  which 
they  contain.  The  carbonic  acid  continually  absorbed  by  the  leaves  is  decom- 
posed by  the  action  of  light  in  those  cells  containing  chlorophyll,  and  the  super- 
fluous oxygen  is  given  out. 

The  oxygen  developed  by  the  plant  may  serve  as  a  measure  of  the  decom- 
position, if  it  is  really,  as  is  generally  assumed,  completely,  or  under  various 
circumstances  in  equal  degree,  separated  by  the  respiratory  organs  of  the 
plant. 

Since  this  decomposition  of  the  carbonic  acid  takes  place  only  in  the  cells 
containing  chlorophyll,  and  in  these  only  under  the  action  of  light,  it  is  sug- 
gestive to  investigate  the  optical  character  of  this  coloring  matter. 

If  we  deprive  a  plant  of  its  chlorophyll  by  treatment  with  water,  alcohol, 
or  ether,  and  examine  the  spectrum  of  the  light  which  passes  through  a  fresh 
concentrated  solution,  it  appears  that  different  parts  of  the  sun's  light  are  more 
or  less  completely  absorbed. 


violet        iiuligo 


The  extreme  red  remains  completely  unchanged.*  but  immediately  behind 


*  The  influence  of  chlorophyll  upon  the  ultra  red  portion  of  the  spectrum  has  not,  so  far  as 
I  know,  been  investigated.    It  is  to  be  desired  that  it  may  soon  be  done. 


88  NOTES  AND  ADDITIONS 

Fraunliofer's  line  B,  Fig.  7,  we  have  a  black  absorption  strip  which  Hagenbach 
denotes  by  /.  This  strip  is  pretty  sharply  defined,  and  extends  beyond  line  C. 
About  at  its  middle  there  is  a  light  portion.  A  second  aljsorption  strip  (//  of 
Hagenbach)  occurs  nearly  in  the  middle  between  G  and  D  ;  a  third  {III.),  a 
little  behind  D  ;  a  fourth  (IV),  in  the  green,  just  before  E.  These  strips  are, 
however,  much  less  dark  than  the  first  in  the  red. 

From  the  middle,  between  E  and  O,  on,  almost  the  rest  of  the  entire  spec- 
trum is  uniformly  absorbed.  The  Figure  shows  the  absorption  spectrum  of 
chlorophyll,  and  indicates  the  position  of  Fraunliofer's  lines. 

We  see  from  the  Figure  that  the  yellow  red  rays,  certain  parts  of  the  yellow, 
and,  in  less  degree,  the  green  and  the  indigo  blue  and  violet  portions  of  (he 
spectrum,  are  almost  completely  cut  off  by  the  chlorophyll. 

The  absorption  spectrum  of  the  solid  chlorophyll,  as  found  in  the  leaf,  agrees 
in  number  and  arrangement  of  the  strips  with  that  of  the  solution,*  and  that 
of  the  leaves  of  living  plants  does  not  deviate. 

Slight  differences  in  the  absorption  phenomena  in  actual  leaf  organs  and  in 
solutions,  are  explained,  as  Melde  suspected  and  Gerland  has  proved,  by  the 
diminution  of  the  light  by  the  other  contents  of  the  cells  and  the  tissues  of  the 
leaves. 

Only  the  absorbed  rays  can  cause  the  processes  of  decomposition  and  assim- 
ilation in  plants,  and  those  rays  not  absorbed  are  of  comparatively  indifferent 
effect. 

If  we  inquire  now  which  of  these  rays  has  the  greatest  effect  upon  chemi- 
cal processes,  we  must  evidently  seek  them  among  those  having  the  greatest 
mechanical  energy. 

This  energy  has  no  connection  with  the  subjective  sensation  of  light  inten- 
sity, for  our  eyes  may  indeed  decide  whether  a  given  red  is  lighter  or  darker 
than  the  red  from  another  source,  but  they  cannot  compare  the  light  rays  of 
two  different  pure  colors  of  the  spectrum. 

The  best  measure  of  the  mechanical  energy  of  a  given  color  is,  as  Lommel 
has  remarked,  the  heat  effect,  when  we  assume  that  the  entire  energy  of 
a  ray  absorbed  by  a  sootcovered  thermopile  is  completely  transformed  into 
heat. 

If  this  is  the  case,  then  the  heat  curve  of  the  solar  spectrum  is  that  which 
expresses  the  energy  of  the  various  rays. 

The  heat  effect  of  the  violet  and  blue  rays  is  very  slight,  and  that  of  the  red 
and  ultra  red  very  large. 

In  the  Figure  the  curved  line  is  the  heat  curve  of  the  solar  spectrum. 

If  we  compare  it  with  the  absorption  spectrum  of  chlorophyll,  we  see  that 
the  yellow  rays  especially  must  furnish  the  energy  required  for  the  assimila- 
tion of  the  carbonic  acid,  for  these  are  the  most  completely  absorbed,  and  pos- 
sess the  greatest  mechanical  energy. 

From  the  preceding  we  can  tell  beforehand  what  influence  the  different 
parts  of  the  spectrum — that  is,  the  color  of  the  acting  light — will  have  upon  the 
activity  of  the  chlorophyll. 

If  we  bring  green  plants  into  such  portions  of  the  solar  spectrum  f  as 
are  not  absorbed  by  the  chlorophyll,  there  can  be  no  decomposition  of  car- 
bonic acid.  If  we  allow  all  that  light  to  act,  corresponding  to  absorption 
strip  /,  there  is  a  very  active  decomposition  of  carbonic  acid  and  development 
of  oxygen. 

The  activity  of  assimilation  must  then  be  little  less  than  one-half  of  that 
for  free  exposure,  if  parts  of  the  ultra  red  portion  of  the  spectrum  are  also 
absorbed  by  chlorophyll. 

'The  amount  of  heat  obtained  by  the  absorption  of  strip  /is  about  equal  to 

*  Lommel  (Pogs:.  .\\m.  vol.  cxliii.,  p.  579)  is  indeed  not  entirely  of  this  opinion,  bnt  Garland's 
objections  (Pogg.  Ann.  vol.  cxliii.,  p.  605)  to  Lommers  views  appear  to  me  well  taken.  Also,  we 
find  here,  that  J.  IMliller's  donbt,  whither  the  spectrum  of  green  leaves  agrees  with  that  of  chloro- 
phyll, i-esrs  only  npon  ob>^i.Tvations  made  under  unfavorable  conditions. 

+  In  order  to  obtain  sufficient  intensitj',  a  concave  mirror  must  be  used  instead  of  the  plane 
mirror  of  the  heliostat,  and  caie  must  be  taken,  by  the  use  of  rock-salt  prisms  and  lenses,  thst  as 
little  heat  as  possible  may  be  absorbed  by  the  apparatus. 


TO    THE  LECTURES.  89 

the  total  beat  obtained  by  tlie  absorption  of  the  entire  portion  of  the  spectrum 
from  between  F  and  G. 

Smaller  maxima  of  carbonic  acid  decomposition  must  occur  when  the  plant 
is  brought  into  strips  II,  III,  IV. 

In  that  part  of  the  spectrum  from  between  F  and  G  on,  there  will  be 
feeble  decomposition. 

If,  by  suitable  means,  we  separated  all  the  colors  from  between  F  and  G 
to  the  red  end  of  the  spectrum,  and  allow  the  others  to  act  upon  the  plant, 
the  assimilation,  assuming  that  the  ultra  red  rays  are  of  no  effect,  will  be  only 
half  as  great  as  for  free  exposure. 

These  assumptions,  deduced  from  a  comparison  of  the  absorption  spectrum 
of  chlorophyll  with  the  heat  spectrum  of  the  sun,  are  confirmed  by  experiment. 

Draper  has  found  the  carbonic  acid  decomposition  greatest  in  the  yellow 
red  of  the  actual  spectrum,  and  his  result  has  been  confirmed  later  by  the 
experiments  of  Sachs,  Prillieux,  A.  Mayer,  Pfeflfer,  and  Bai-inetzky. 

Most  of  these  investigators  have  worked  with  colored  glasses  and  solutions, 
and  not  with  the  actual  spectrum,  and  their  results  therefore  cannot  be  at  once 
made  use  of. 

If  useful  results  are  expected  witli  colored  glasses  and  solutions,  we  must 
not  only  determine,  as  was  done  by  Sachs,  the  absorption  spectrum  of  the  glass 
or  solution,  but  also,  by  special  photometric  measurements,  what  portions  of 
the  parts  of  the  spectrum  absorbed  by  the  chlorophyll  are  still  effective  in  the 
spectrum  of  the  glass  or  solution. 

When  this  is  done,  the  results  will  undoubtedly  be  in  agreement  with  theory. 

Whether  by  the  fluorescent  properties  of  chlorophyll  any  change  will  be 
caused,  cannot  be  decided  without  further  information;  but  it  does  not,  in  view 
of  the  theoretical  investigations  of  Lommel ,  appear  probable. 


NOTE  32.— (Page  51.) 

VIEWS   OF   MAYER   UPOK   THE   PHEN0MEN0:N'   OE   THE   TIDES. 

It  may  not  be  superfluous  to  add  a  few  words  upon  an  interesting  astronomi- 
cal application  of  the  theory,  the  first  suggestion  of  which  is  due  to  Mayer. 

We  know  that,  by  reason  of  the  combined  action  of  the  sun  and  moon,  two 
waves  originate  at  opposite  points  upon  the  sea  and  traverse  the  earth,  caus- 
ing the  phenomenon  of  the  tide.  When  the  tidal  wave  meets  the  coast  and 
shores,  it  causes  currents  and  counter-currents,  which  cannot  exist  without 
friction  and  development  of  heat. 

We  have  thus  upon  the  surface  of  our  planet  a  generation  of  heat.  But 
the  total  living  force  (energy)  of  the  planet  cannot  be  increased  by  the  mutual 
action  of  its  difl'erent  parts,  and  hence  this  apparent  creation  of  heat  can  only 
be  a  transformation  of  other  living  force  (kinetic  energy)  into  the  living  force 
of  heat  (caloric  energy). 

The  ebb  and  flow  of  the  tides  diminish,  thereiore,  incessantly  the  living 
force  (kinetic  energy)  of  the  earth.  Probably,  both  the  velocity  of  rotation  and 
the  rectilinear  rotation  diminish  together  ;  that  is,  the  length  of  the  sidereal 
day  increases,  and  the  major  axis  of  the  earth's  orbit  diminishes. 

Similarly  to  the  tides  upon  the  surface  of  the  ocean,  Falb  *  has  assumed  a 
corresponding  tide  upon  the  surface  of  the  fluid  interior  of  the  earth,  and  de- 
duces from  this  assumption  the  regular  return  of  earthquakes  and  volcanic 
eruptions. 

If  the  solid  earth  crust  and  the  fluid  interior  are  without  any  intermediate 
space  between  them,  such  a  tide  could  not  form,  but  the  tendency  to  its  forma- 
tion would  be  indicated  by  an  increase  of  pressure  of  the  fluid  contents  against 
the  crust. 

*  Falb,  Grundziigc  zu  eine  Theorie  der  Erdbeben  und  Vulcanansbriicbe.    Graz,  18~1. 


90  NOTES  AND  ADDITIONS 

If,  howevei",  such  tides  actually  occur  in  the  earth's  interior,  they  can 
effects  similar  to  those  of  tlie  tides  on  the  surface. 

It  is,  indeed,  true  that  the  change  in  length  of  the  day,  and  the  diminution 
of  the  axis  of  the  earth's  orbit,  are  so  small  that  they  would  be  imperceptible 
even  in  the  course  of  centuries.*  Still,  from  a  theoretical  standpoint,  these 
conclusions  are  none  the  less  interesting. 

Such  tides  have  certainly  occurred  upon  the  fluid  masses  of  the  planets, 
while  they  were  still  in  a  fluid  condition. 

It  may  be  that  such  tides,  caused  by  the  action  of  the  earth  upon  the  moon, 
have  deprived  it  of  its  axial  rotation,  while  it  was  still  in  the  fluid  condition, 
and  this  may  be  the  reason  why  now  it  always  presents  to  us  the  same  side. 


NOTE  33.— (Page  54.) 

UPOK   A   EEMAKK   OF   SEGUIN   COi^CERlSriKG   THE   STEAM   EKGINE. 

In  order  to  prove  that  during  the  action  of  a  steam  engine  heat  is  neces- 
sarily used,  Seguin  remarks  that  if  all  the  heat  taken  from  the  boiler  were 
found  in  the  condenser,  this  amount  of  heat  would  be  sufficient  to  repeat  the 
same  action  indefinitely,  provided  that  it  were  possible  to  concentrate  the  heat 
contained  in  the  condenser  water,  so  that  with  it  the  fifteenth  part  of  its  mass 
could  be  heated  to  100%  and  then  converted  into  saturated  steam  at  this  tem- 
perature ;  which  is  in  entire  agreement  with  theory. 

We  could  thus  obtain,  by  means  of  a  finite  amount  of  heat,  an  indefinite 
continuance  of  motion,  which  is  neither  probable  nor  in  accordance  with  sound 
logic. 

This  argument  is  not  completely  satisfactory,  because  the  concentration  of 
heat  assumed  by  Seguin  implies  an  equivalent  expenditure  of  work  or  heat. 
According  to  this  argument,  a  body  must  be  brought  to  a  temperature  of  100' 
by  heat  taken  from  another  body  at  40'. 

We  have  seen  in  the  preceding  presentation  under  what  conditions  this  is 
possible. 


POSTSCEIPT    OF   VERDET   TO   THE   N"OTES,    JULY   16,    1863. 

While  in  press,  I  received  the  "  L'exposition  analytique  et  experimentale  de 
la  theorie  niecanique  de  la  chaleur,"  by  Him,  forwarded  by  the  author  to  the 
Academy  at  the  session  of  July  7,  1862.  In  this  work  Hirn  recognizes  the 
error  of  his  earlier  conclusions,  and  gives  the  explanation  of  the  peculiar  re- 
sults furnished  by  his  experiments  upon  engines  without  expansion.  Our 
criticisms  upon  this  savant  (page  14)  are,  therefore,  rendered  inapplicable. 


NOTE  34.— (Page  54.) 


THE    DEPEN"DEKCE    OF    THE    COLOR    OF    VENOUS    BLOOD    UPON"    THE 
TEMPERATURE. 

For  a  fuller  understanding,  it  may  be  well  to  add  some  information  as  to 
the  anatomical  and  physiological  relations  which  exist  in  the  higher  animal 
organisms. 

■^  Mayer  estimates  that  the  leii'jth  of  the  clay  would  be  increased  by  the  action  of  the  tides 
one-sixteenth  of  a  second  in  ;3,500  years. 


TO    THE  LECTURES.  91 

The  blood-vessels  are  of  three  classes. 

1.  The  arteries,  which  carry  the  blood  from  the  heart  to  all  parts  of  the 
body. 

2.  The  veins,  which  carry  the  blood  back  again  to  the  heart. 

3.  The  capillaries,  which  unite  the  extreme  branches  of  the  arteries  and 
the  veins. 

In  these  blood-vessels  the  blood  circulates  incessantly,  so  long  as  life  exists. 
The  apparatus  which  causes  the  motion  of  the  blood  is  the  heart.  It  is  divided 
into  two  parts,  the  right  and  left,  by  an  impervious  partition.  Each  half 
consists  of  a  ventricle  and  auricle.  Each  ventricle  is  connected  by  valves 
with  its  auricle.  The  sides  of  the  heart,  especially  the  left,  are  formed  of 
powerful  muscles.  From  the  left  ventricle,  at  each  contraction  (systole),  a 
portion  of  the  bright  blood  in  it  is  forced  out  into  the  aorta,  and  thence  into 
all  the  other  arteries.  Since  the  arteries  are  already  filled  with  blood,  they 
are  expanded  by  the  entrance  of  the  new  blood  When  the  pressure  from  the 
heart  ceases,  and  they  contract  (diastole),  valves  at  the  entrance  of  the  aorta 
prevent  the  return  of  blood  to  the  heart.  The  contraction  of  the  elastic  walls 
of  the  arteries  forces  the  blood  forward,  through  their  ramifications,  into  the 
capillaries,  and  from  these  into  the  veins.  The  blood  thus  proceeding  from  the 
heart  is  called  arterial,  and  is  charged  with  oxygen. 

In  the  capillaries,  which  permeate  all  the  tissues,  this  oxygen  combines 
with  the  carbon  and  hydrogen  of  the  food.  'J'he  heat  generated  by  this  and 
other  chemical  processes  is  one  source  of  the  animal  heat  of  living  organisms. 

By  this  combustion  carbonic  acid  is  formed,  which  is  held  in  solution  in  the 
blood,  and  gives  it  a  darii  color. 

Ths  dark  colored  blood,  thus  deprived  of  its  oxygen  and  permeated  with 
carbonic  a.-id,  is  called  venous  blood.  The  veins  carry  this  blood,  and  with  it 
many  products  of  digestion,  which  are  likewise  taken  up  by  it  in  the  capillaries, 
to  tlie  hep.rt.  It  enters  there  the  right  auricle,  and  has  thus  made  the  "  greater 
circuit."  When  the  contracticju  ceases  and  the  heart  expands,  the  l)lood  enters 
from  the  auricles  the  ventricles,  and  thus  the  venous  blood  passes  from  the 
right  auricle  into  the  right  ventricle.  From  this  it  is  expelled  by  the  next 
contraction,  and  forced  into  the  blood-vessels  of  the  lungs.  These  branch  out 
in  the  lungs  into  extremely  fine  capillary  tubes,  which  surround  like  a  network 
the  countless  ramifications  of  the  air  passages. 

Through  the  very  thin  sides  which  separate  the  blood  from  the  air  passages, 
there  is  an  interchange  of  gases.  The  carbonic  acid  in  the  blood  is  given  out, 
and  the  oxygen  of  the  air  is  absorbed. 

The  blood  thus  retakes  its  bright  red  color,  and  becomes  again  arterial. 

From  the  capillary  vessels  of  the  lungs,  this  arterial  blood  passes  by  four 
veins  back  to  the  heart,  and  enters  the  left  auricle.  From  this,  at  the  next 
diastole,  it  passes  into  the  left  ventricle,  and  begins  at  the  next  systole  its 
course  anew. 

The  passage  of  the  blood  from  the  right  ventricle,  through  the  capillary 
system  of  the  lungs  back  to  the  left  auricle,  we  call  the  lesser  circuit. 

The  human  body  is  thus  comparable  to  a  steam  engine.  The  nourishment 
is  the  fuel,  from  the  combustion  of  which  arises  the  increased  temperature  and 
power  of  both.  In  both  cases  the  oxygen  necessary  for  combustion  is  taken 
from  the  air.  To  the  boiler  correspond  the  capillary  vessels.  As  chimney  for 
the  discharge  of  the  carbonic  acid  formed  by  combustion,  and  at  the  same  time 
as  grate  through  which  the  air  enters,  we  have  the  lungs  and  wind-pipe. 

As  already  noticed,  the  heat  generated  in  the  body  by  combustion  of  the 
nourishment,  serves  a  double  purpose.  A  portion  is  transformed  by  the 
muscles  into  work  ;  another  serves  to  maintain  the  temperature  of  the  body 
constant,  that  is,  to  supply  the  losses  by  radiation,  conduction,  perspiration, 
etc. 

Many  observations  have  shown  that  the  temperature  of  the  human  body  is 
independent  of  the  outer  temperature,  and  in  a  healthy  condition  about  37°  C. 

When  the  outer  temperature  is  low,  the  loss  of  heat  must  then  be  greater, 
and  more  heat  must  be  produced  to  cover  this  loss  than  when  the  outer 
temperature  is  higher.     For  low  outer  temperature,  therefore,  more  nourish- 


92  NOTES  AND  ADDITIONS 

ment  must  be  burned,  aud  in  regions  where  the  outer  temi3erature  is  liiglisr, 
less. 

When  more  nourishment  is  burned,  more  oxygen  must  be  inspired,  and 
more  carbonic  acid  formed.  Hence  in  cold  climates  the  venous  blood  must 
contain  more  carbonic  acid  than  in  the  warm,  and  the  arterial  more  oxygen. 

The  difference  in  color  between  venous  and  arterial  blood  depends  upon 
the  greater  amount  of  carbonic  acid  in  the  one,  and  of  oxygen  in  the  other. 
There  must,  therefore,  be  a  relation  between  the  difference  of  color  and  the 
amount  of  nourishment  consumed,  or  between  this  difference  of  color  and  the 
animal  heat. 

Hence  the  color  of  the  venous  blood  must  vary  with  the  outer  temperature. 

In  cold  climates  the  venous  blood  is  darker  than  in  warmer.  Hence  it  was 
that  Mayer  found  in  Java  the  venous  blood  much  redder  than  in  Europe. 

This  observation  furnished  him  the  starting-point  for  his  discovery  of  the 
fundamental  principle  of  the  mechanical  equivalent  of  heat.* 


*'THE   ENTEOPY   OF   THE   WORLD   TEN^DS   TOWARD   A    MAXIMUM."  f 

In  Chapter  IV.  of  the  second  Lecture  we  have  called  attention  to  an  impor- 
tant law :  the  amount  of  heat  which  in  a  perfect  heat  engine  is  transformed 
into  work,  and  the  amount  which  is  simultaneously  transferred  from  a  hot 
body  (the  furnace)  to  a  colder  body  (the  condenser),  stand  in  a  constant  rela- 
tion. 

This  principle  is  expressed  by  the  formula 

(«,  -  h) 

or,  denoting  1  +  a^,by  Ti,  the  absolute  temperature, 

.     q-q'  _T.-T, 
q       -        T,        ■ 

Her?  q  is  the  total  heat  expenditure  of  the  warmer  body,  of  the  boiler, 
whose  temperature  is  7'i,  and  q'  is  that  part  of  this  heat  which  is  transferred 
to  the  colder  body,  to  the  condenser,  whose  absolute  temperatiire  is  Tq. 

Subtracting  both  members  of  the  above  equation  from  1,  we  have 


which  may  be  written 


^  =  ^'1         or        -^-^ 
q        T,  '  '        r,  ~  T^ 


A-|;  =  o (.). 


This  principle  admits  of  considerable  extension,  and  holds  good  in  some- 
what changed  form,  not  only  for  the  processes  accomplished  in  heat  engines, 
but  for  all  processes  by  which  heat  is  transformed  into  work,  or,  inversely, 
work  into  heat. 

Since  the  change  of  heat  into  work,  and  likewise  the  transfer  of  heat  of 
high  temperature  into  heat  of  low  temperature,  is  a  transformation,  this  prin- 

*  See  Mayer,  Mechanik  der  Warmrt,  1867.  p.  230. 

+  Seo  Claiisius,  ueber  den  zweiten  Hauptsarz  dcr  mechanischen  Wiirmctheorie,  Braunschweig, 


TO   THE  LECTURES.  93 

ciple  is  called  by  Clausius,  the  law  of  tlie  equivalence  of  transformations,  and 
is  regarded  as  tlie  second  general  law  of  thermodynamics. 

It  holds  good,  as  we  see,  only  for  cycle  processes  which  are  reversible,  as  in 
that  of  the  perfect  heat  engine. 

If  the  cycle  process  is  more  complex,  the  equation  above  is  more  generally 


(I) 


There  are,  however,  transformations  of  other  forms  of  force  into  each  other, 
and  all  follow  this  law. 

Thus,  for  example,  heat  changes  the  arrangement  of  the  molecules,  while 
overcoming  outer  forces  and  the  action  of  the  molecular  forces  ;  while,  there- 
fore, it  performs  work. 

When  the  action  of  the  heat  is  sufficiently  powerful,  solid  bodies  become 
liquid,  and  liquids  are  changed  into  gase.^;  ;  therefore  the  state  of  aggregation  is 
changed.  Clausius  calls  this  action  of  the  heat  "  disgregation,"  and  expresses 
the  ])lienomenon  by  the  words,  "  heat  increases  the  disgregation  of  bodies." 

An  increase  of  disgregation  corresponds,  then,  to  a  change  of  heat  into 
work,  and  a  decrease  to  a  transformation  of  woi-k  into  heat.  There  must, 
therefore,  exist  between  the  decrease  of  disgregation  and  such  transformation 
a  causal  relation. 

That  the  consideration  of  this  disgregation  conducts  us  to  the  preceding 
equation,  may  be  seen  from  the  following  example: 

Let  us  consider  a  quantity  of  gas  which  has  the  temperature  t,  volume  v, 
and  pressure  p.  Since,  under  such  conditions,  the  mean  distances  of  the  mole- 
cules is  determinate,  the  disgregation,  which  measures  the  distribution  of  the 
gas  particles,  is  determinate.  We  denote  this  by  Z.  If  we  now  allow  the  gas 
to  expand,  or  compress  it,  under  constant  outer  pressure,  without  change  of 
temperature,  the  work  performed  or  expended  \sp{v^  —  v),  where  «,  is  the 
new  volume. 

The  corresponding  amount  of  heat  q,  absorbed  or  set  free,  is 

,=  l<^ (2). 

Let  the  disgregation  in  this  new  condition  be  Z^,  then  the  change  of  disgre- 
gation is 

Z,  -Z. 

Let  us  now  consider  an  equal  amount  of  the  same  gas  which  has  for  another 
temperature  t, ,  the  same  volume  v,  and  hence  another  pressure  p,. 

The  pressures  p  and  p^,  according  to  the  laws  of  the  expansion  of  gases, 
are  in  the  relation 

p    _  1  +  g- 1 

The  disgregation  of  the  new  gas  mass  is  evidently  the  same,  viz.,  Z,  since 
the  mean  distance  of  the  molecules  is  the  same. 

If  this  gas  takes  the  new  volume  «,,  the  change  of  disgregation  is  neces- 
sarily the  same,  viz.,  Z^  —  Z. 

The  work  obtained  or  expended  is  however  evidently  different,  viz., 

^1  (»,  —  V), 

and  the  heat  g,,  which  is  absorbed  or  set  free,  if  the  temperature  of  the  gas 
does  not  change,  is 

S.=«^V^ (8,. 


Ql  NOTES  AND  ADDITIONS 

From  equations  (2)  and  (3)  we  have 

—  and  — 


Pi  J  P  J 

The  left  sides  are  evidently  alike.     If  we  take  account  also  of  the  equation 

p_  _  Ij- n^j  _   T^ 
p[  "~  iT'cVT^  ~  T,' 

we  have,  when  Tand  T^  are  the  absolute  temperatures, 

|f=f. (4), 

an  equation  which  corresponds  perfectly  witli  (1). 

We  see,  therefore,  that  the  amounts  of  heat  necessary  to  cause  the  same 
change  of  disgregation  are  inversely  as  the  absolute  temperatures  at  which 
these  amounts  of  heat  are  transformed. 

The  equivalent  value  of  the  heat  corresponding  to  a  determinate  change  of 
disgregation  is,  therefore,  obtained  by  dividing  the  heat  necessary  for  this 
change  by  the  absolute  temperature. 

These  two  examples,  viz.,  the  theory  of  macliines  already  alluded  to,  and 
the  law  of  disgregation  change  here  laid  down,  may  suffice,  if  not  to  prove 
rigidly,  at  least  to  make  intelligible  the  second  law  of  thermo-dynamics. 

We  have  now,  in  the  course  of  our  considerations,  become  acquainted  with 
three  kinds  of  changes,  viz,,  the  change  of  heat  into  work,  or,  inversely,  the 
change  of  heat  at  a  higher  temperature  into  heat  at  a  lower,  and,  finally,  dis- 
gregation changes. 

Every  transformation  of  one  kind  corresponds  always  to  a  certain  amount 
of  another,  and  we  can  therefore  say  that,  iu  every  process,  a  change  of  one 
sort  answers  always  to  a  corresponding  change  of  another. 

Entirely  analogous  relations  can  be  stated  for  every  transformation  of  one 
kind  of  force  into  another. 

If  we  assume,  temporarily,  a  unit  for  two  equivalent  transformations,  we 
can  set  them  equal. 

Since  every  equation  of  tlje  form 

«,  —  V2 
can  be  put  in  the  form 

Vi  —  V,  —  0, 

this  principle  can  be  expressed  as  follows  : 

In  every  process  the  algebraic  sum  of  the  transformations  is  zero. 

It  is  necessary,  however,  to  call  attention  to  the  limitation  which  in  the  one 
case  is  expressly  made,  and  in  the  other  is  fulfilled,  viz.,  that  the  process  in 
question  must  be  reversible. 

Keeping  this  limitation,  we  have  the  second  law  of  thermo-dynamics  in  the 
form  as  given  by  Clausius. 

"  In  every  process,  however  complicated,  in  which  one  or  more  bodies 
undergo  reversible  changes,  the  algebraic  sum  of  all  the  transformations  must 
be  zero." 

The  second  law  is,  therefore,  well  called  the  law  of  the  equivalence  of 
transformations,  while  the  first  is  that  of  the  equivalence  of  work  and  heat. 


We  shall  next  seek  to  make  it  evident,  by  further  examples,  that  it  is  really 
necessary  to  introduce  the  limitation  that  the  second  law  holds  only  for  reversi- 
ble transformations. 


TO   THE  LECTVBE8.  95 

In  the  example  already  noticed,  in  Avhicli  M'e  consider  the  change  of  dis- 
gregation  of  a  gas,  it  was  always  assumed  that  the  pressure  remained  un- 
changed, or,  better,  that  the  outer  pressure  differed,  from  the  tension  of  the  gas 
only  by  an  infinitely  small  amount.  Under  this  assumption  it  is  possible  to 
again  compress  the  gas  by  the  same  outer  pressure,  and  bring  it  back  to  its 
original  condition. 

The  gas  then  passes  through  all  the  changes  which  it  experienced  during 
expansion,  but  in  reverse  order. 

The  gas,  however,  may  experience  the  same  changes  of  volume  and  dis- 
gregatiofl  in  another  manner. 

If,  thus,  we  connect  the  vessel  containing  the  gas,  whose  volume  is  v,  with 
another  whose  v^olume  is  «,  —  w,  which  is  exhausted  of  air,  and  suddenly 
open  the  communication,  the  gas  will  enter  the  empty  vessel  until  there  is  the 
same  pressure  in  both.  The  volume  of  the  gas  is  now  «,  ;  the  change  of  dis- 
gregation  is  the  same  as  in  the  previous  case. 

From  the  experiments  of  Joule  we  know  that  during  such  a  change  of 
volume  there  is  neither  change  of  temperature  nor  work  performed  by  the  gas. 

But  the  gas  cannot  be  compressed  back  to  its  original  condition  without  an 
expenditure  of  work  and  production  of  heat.     The  process  is  not  reversible. 

If  the  gas  is  compressed,  its  disgregation  therefore  diminished,  we  must 
have  work  transformed  into  heat  ;  but,  as  we  have  seen,  the  disgregation  may 
he  increased  without  an  equivalent  transformation  of  work  into  heat  or  heat 
into  work. 

If,  now,  we  call  the  transformation  of  work  into  heat  and  increase  of  dis- 
gregation positive,  and  the  change  of  heat  into  work  and  decrease  of  disgrega- 
tion negative  transformations,  we  see  that  decrease  of  disgregation,  that  is,  a 
negative  change,  cannot  occur  without  a  simultaneotts  positive  transformation; 
but,  on  the  other  hand,  increase  of  disgregation,  or  a  positive  change,  can 
sometimes  occur  without  a  negative  transformation. 

Let  us  consider  now  other  modes  of  transformation.  When  heat  is  trans- 
formed into  work  there  is  always  a  simultaneous  increase  of  disgregation,  or, 
as  in  the  cycle  processes  of  engines,  heat  passes  from  a  hot  to  a  colder  body. 
If  we  call  the  transfer  from  the  hot  to  the  cold  body  positive,  and  the  trans- 
formation of  heat  into  work  negative,  we  can  say,  since  there  is  i-o  example  in 
which  this  negative  transformation  occurs  without  a  corresponding  positive 
one,  that  the  negative  transformation  of  heat  into  work  is  necessarily  connected 
with  a  simultaneous  positive  transformation. 

The  positive  change  of  work  into  heat  can,  however,  as  many  examples 
shov/,  occur  without  a  corresponding  simultaneous  negative  transformation. 
Thus,  for  example,  in  friction,  resistance  of  air,  and,  in  short,  most  prejudicial 
resistances,  there  is  a  change  of  work  into  heat,  without  simultaneous  changes 
of  disgregation,  transfer  of  heat  from  higher  to  lower  temperature,  etc.,  neces- 
sarily occurring. 

Here  also,  therefore,  negative  transformation  of  heat  into  work  cannot 
occur  without  a  simultaneous  positive  transformation,  but  positive  transforma- 
tion of  work  into  heat  can. 

The  third  mode  of  transformation  considered — viz.,  the  transfer  of  heat 
from  one  body  to  another,  or  the  change  of  a  quantity  of  heat  Q  at  the  tem- 
perature T,  into  the  quantity  Q  at  the  temperature  T' — also  confirms  this  law. 

As  is  known,  there  is  a  natural  tendency  of  heat  to  pass  from  a  warmer 
to  a  colder  body;  and  this  process  occurs  in  radiation  and  conduction  without 
simultaneously  giving  rise  to  another.  ^ 

On  the  other  hand,  a  process  opposed  to  this  natural  tendency  of  heat,  the 
transfer  of  heat  from  a  colder  to  a  warmer  body,  can  only  occur  when  there  is 
a  simultaneous  change  of  work  into  heat,*  or  an  increase  of  disgregation. f 

If  we  call,  as  already  indicated,  the  transfer  of  heat  from  hot  to  cold  body 

*  We  may  recall  the  example  on  page  15  of  a  steam  engine  compelled  \>y  an  onter  force  to 
reverse  its  ordinarj'  action. 

t  As  illnstration,  a  hot  gasof  temperature  A"  must  be  compressed  by  tlie  expansion  of  a 
colder  solid  body  which  is  heated  from  A  to  A',  when  both  A  and  A'  are  les-=  than  A". 


96  NOTES  AND  ADDITIONS 

positive,  and  from  cold  to  hot  negative,  we  may  conclude  from  all  tlie 
discussed,  that 

Negative  transformations  can  only  occur  wlien  compensated  by  positive,  hut 
positive  may  occur  loitliout  negcttive.  Uncompensated  transformation's  can  there- 
fore only  he  positive. 

This  principle  allows  of  an  interesting-  natural  application.  We  have  seen 
that  there  is  a  general  tendency  in  nature  to  increase  of  disgregation,  to  trans- 
form work  into  heat,  and  to  level  heat  differences.  This  is  indeed  but  the 
result  of  the  principle  that  uncompensated  transformations  can  only  be  positive. 

This  tendency  is  denominated  the  "  dissipation  of  energy." 

The  case  of  a  natural  transformation  which  is  perfectly  reversible  is  a 
limiting  case  which  seldom  or  never  occurs.  It  is,  hence,  the  tendency  of  the 
positive  transformations  to  accumulate.  The  heat  of  a  body  which  can  no 
longer  be  transferred  to  a  colder  body  must  remain  heat,  and  can  serve  no 
longer  for  production  of  work,  can  no  longer  be  transformed  into  other  forms 
of  action. 

The  amount  of  this  untransformable  heat  must,  hence,  always  increase,  since 
it  is  continually  added  to  by  the  uncompensated  positive  transformations. 

The  consequence  is  that  the  world  tends  toward  a  final  condition  in  which 
all  its  forms  of  energy  will  be  transformed  into  heat  of  uniform  temperature, 
which  can  no  more  be  transformed. 

When  this  condition  is  attained  all  nature  will  be,  and  must  remain,  dead. 

Thomson  has,  with  rare  acuteness,  drawn  the  boldest  consequences  from 
these  conclusions,*  and  expressed  them  as  follows  : 

"  1.  There  is  in  nature  a  universal  tendency  to  the  dissipation  of  mechani- 
cal energy. 

"  2.  A  restoration  of  mechanical  energy  (negative  change)  without  more  than 
an  equivalent  of  dissipation,  is  impossible  by  inanimate  material  processes,  and 
will  probably  never  be  attained  by  organized  matter,  whether  endowed  with 
vegetable  life  or  with  consciousness  and  will. 

"  3.  Within  a  finite  past  period  the  earth  must  have  been  uninhabitable,  and 
within  some  finite  future  period  the  earth  must  become  again  uninhabitable  for 
men,  beasts,  and  plants  as  now  constituted.  It  may  be  then  that  processes 
may  have  existed  or  will  exist,  which  are  contrary  to  those  natural  laws  which 
at  present  rule  the  world." 

The  boldness  and  scope  of  the  conclusions  drawn  by  Thomson  from  the  sim- 
ple equations  which  express  the  second  law  of  thermo-dynamics,  must  challenge 
admiration. 

The  merit  of  the  first  discovery  of  these  equations,  however,  belongs  to 
Carnot  and  Clausius.  Carnot  discovered  the  second  law  in  its  essentials, 
though  the  form  in  which  he  expressed  it  was  incorrect,  since  he  proceeded 
from  the  false  assumption  of  the  indestructibility  of  heat. 

Later,  Clausius  so  modified  Cavnot's  principle,  that  it  no  longer  contra- 
dicted the  first  law  and  the  principle  of  the  conservation  of  fone,  and  ex- 
pressed it  in  correct  fortn.f  He  showed  that  it  followed  from  the  inherent 
tendency  of  heat  to  equalize  existing  temperature  differences. 

From  the  second  law,  however,  it  follows  that  transformation  of  heat  into 
work  can  never  occur  without  compensation  by  a  transfer  of  heat  from  a 
warmer  to  a  colder  body.  But  already,  at  that  time,  known  phenomena  taught 
that  heat  could  pass  from  a  warmer  to  a  colder  body  without  compensation,  and 
that  work  could  be  transformed  into  heat  without  compensation,  as  in  friction. 

Later,  Clausius  gave  in  another  form  its  most  general  expression  to  the 
law  which  we  have  already  expressed,  by  saying  that  imcompensated  changes 
can  only  be  positive. | 

Clausius  denoted  the  algebraic  sum  of  all  those  changes  which  must  take 

*  Thomson,  Proceed,  of  the  Royal  Soc.  of  Edinburgh,  April,  185-2,  nnd  Phil.  Mag.,  4  Series,  ' 
vol.  iv.,  p.  304,  "  On  a  Universal  Tendency  in  Nature  to  the  Dissipation  of  Mechanical  Energy.' 

t  Clausius,  Pogg.  Ann.  Bd.  79,1850.  Abhandlungen  Bd.  1,  p.  50.  "  Ueber  die  bewegende 
Kraft  der  Warme,"  etc. 

X  Clausius,  "  Ueber  den  zweiten  Hnuptsatz  der  mechanischen  Warmetheorie,"  Braunschweig, 
1867,  p.  17. 


TO   THE  LECTURES.  97 

place,  in  order  to  bring  a  body  iuto  the  condition  in  which  it  is  at  present,  by 
the  name  "entropy." 

Since  now  the  sum  of  the  positive  changes  can  never  be  less  than  the  sum 
of  the  simultaneous  negative  change,  but  in  general  is  greater,  it  follows  that 
the  entropy  of  the  world  is  continually  increasing. 

Clausius  has  formulated  this  in  the  words— "i?te  Entropie  der  Welt  strebt 
iincm  mnximum  zu." 

"  The  Entropy  of  the  world  tends  toward  a  maximum." 


THEEMODYlSrAMICS, 


PAKT    FIRST. 


GENERAL  PEINCIPLES.— AIR  AND  HOT-AIR  ENGINES. 


PART    FIRST. 


CHAPTEK  I. 

GENEEATION  OF  HEAT  BY  MECHANIC.IL  WOKK  AND  THE  EEVEESE. 
— ^DETEEMINATION  OF  THE  MECHANICAL  EQUIVALENT  OF  HEAT 
BY   EXPEEIMENT. 

Heat  Generated  by  Meclianical  Action. — A  great  number  of 
the  phenomena  of  daily  life  prove  to  us  that  heat  can  be  gen- 
erated by  mechanical  work.  If,  for  example,  we  strike  a  piece 
of  metal  repeatedly  with  a  hammer,  the  metal  becomes  heated. 
The  living  force  of  the  hammer,  which  is  here  destroyed  at 
every  stroke,  appears  thus  to  be  transformed  into  heat.  If  we 
rub  two  pieces  of  dry  wood  together  with  sufficient  rapidity, 
and  during  a  sufficiently  long  time,  they  may  even  be  set  on 
fire.  Here,  again,  we  have  heat  generated  by  mechanical  work. 
Especially  well  known  is  the  generation  of  heat  by  axle  fric- 
tion when  the  axle  is  not  well  lubricated.  Here,  again,  the 
work  necessary  for  overcoming  the  friction  generates  heat. 

These  and  many  other  facts  have  led  to  the  question  whether 
mechanical  work  and  heat  are  not  equivalent,  or,  in  other 
words,  whether  by  the  expenditure  of  a  certain  amount  of 
work  we  can  generate  a  certain  fixed  quantity  of  heat. 

Approximate  Determination  of  the  Heat  Generated  hy  Fric- 
tion.— Count  Eumford  seems  to  have  been  the  first  to  en- 
deavor to  answer  this  question  by  experiment,  and,  indeed, 
seems  to  have  been  among  the  first  to  express  the  idea  that 
heat  is  nothing  more  than  a  motion  of  the  molecules  of  a  body. 

In  the  foundry  at  Munich,  of  which  he  was  the  superintend- 
ent, he  caused  a  blunt  drill,  of  about  10,000  pounds  in  weight, 
set  in  motion  by  horses,  to  work  upon  the  bottom  of  a  cannon. 
The  cannon  was  inclosed  in  a  wooden  box  containing    26.6 

101 


102  THEBMODYNAMICS. 

pounds  of  water.  At  the  end  of  2|  hours  the  water  was  heated 
from  0''  to  100'',  or  was  caused  to  boil. 

Although  he  made  use  of  two  horses  for  the  experiment,  he 
was  of  the  opinion  that  the  work  could  have  been  performed  by 
one.  Hence  the  mechanical  work  of  one  horse  for  21  hours 
sufficed  to  heat  26.6  pounds  of  water  through  100^. 

Since,  now,  one  horse  can  raise  in  one  hour  1,980,000  pounds 
one  foot  high,  or  can  perform  a  mechanical  work  of  1,980,000 
foot  lbs.,  in  2i  hours  it  would  perform  1,980,000  x  24  =  4,950,- 
000  foot  lbs.  This  work  heats  26.6  lbs.  of  water  through  100°, 
or  2,660  lbs.  through  1".  Therefore,  to  heat  one  pound  of 
water  one  degree  requires 

4,950,000      -  „^,  ,     ,  „ 
o^^A     =  1'721  foot  lbs. 


Let  us  reduce  this  result  to  French  measures,  which  are 
almost  universally  used  now  in  science,  and  which  we  shall 
always  use  hereafter  unless  the  contrary  is  specially  stated. 

One  pound  equals  0.4536  kilogram,  and  one  foot  equals 
0.3048  meter ;  hence  1  foot  lb.  =  0.4536  x  0.3048  =  0.13826  me- 
ter-kilograms, and  1,721  foot  lbs.  =  1,721  x  0.13826  =  237.945 
meter-kilograms. 

This  result  is  evidently  too  large,*  as  we  have  taken  no  ac- 
count of  the  heat  lost  by  radiation  and  conduction.  Thus  it  is 
plain  that  by  the  same  mechanical  work  we  could  have  heated 
a  greater  quantity  of  water,  or  that  for  the  same  amount  of 
water  less  work  would' have  been  necessary  if  all  the  heat  gen- 
erated had  gone  to  raise  the  temperature. 

Experiment  of  Davy. — After  Kumford  we  find  Sir  Humphry 
Davy  announcing  clearly  that  heat  is  not  a  matter  transmitted 
from  one  body  to  another,  as  was  held  by  most  physicists ;  but 
that  it  consisted,  most  probably,  in  a  rapid  motion  of  the  par- 
ticles of  a  body. 

He  showed  that  two  pieces  of  ice,  when  rubbed  together,  were 
converted  into  water,  although  no  heat  was  imparted  to  them 
by  exterior  bodies.     But  we  know  that  ice  requires  for  melting 

*  [Note  that  this  result  corresponds  to  one  pound  of  water  hented  one  degree.  One  kilogram, 
or  2.3  lbs.,  will  require  2.2  as  much  work,  or  considerably  more  than  424  nieter-kilograms.] 


GENEBATION  OF  HEAT.  103 

a  large  amount  of  heat ;  for  instance,  to  convert  one  pound  of 
ice  at  0"  into  water  at  0'  requires  no  lesslieat  than  to  raise  the 
same  quantity  of  water  from  0°  to  about  80°.  This  considera- 
ble amount  of  heat  must  therefore,  in  the  above  experiment, 
have  been  generated  by  the  friction  of  the  pieces  of  ice,  and 
hence  Davy  concluded  that  heat  must  be  a  kind  of  motion. 

In  order  to  confirm  still  further  the  truth  of  this  view,  he 
made  the  following  experiment :  He  placed  under  the  receiver 
of  an  air-pump  a  clock-work,  resting  upon  a  piece  of  ice,  in 
the  upper  surface  of  which  there  was  a  cavity  filled  with  water. 
The  clock-work  caused  a  toothed  wheel  to  rub  against  a  sur- 
face of  metal  coated  with  wax.  After  the  exhaustion  of  the 
air  the  clock-work  was  set  in  motion,  and  the  wax  was  melted, 
while,  the  water  in  the  ice  cavity  remained  fluid.  The  melting 
of  the  wax,  therefore,  was  not  due  to  heat  obtained  from  the 
water  nor  from  the  clock-work ;  for,  in  the  first  case,  the  water 
would  have  been  at  least  partially  frozen,  and  in  the  other, 
heat  would  have  been  imparted  also  to  the  ice,  and  a  por- 
tion of  it  melted. 

Mayer,  the  Founder  of  the  Theory. — Although  the  savans 
just  mentioned  were  of  the  opinion  that  heat  must  be  a  kind 
of  motion,  and  that  a  definite  expenditure  of  work  must  gener- 
ate an  equivalent  amount  of  heat,  to  Dr.  Mayer  of  Heilbronn 
belongs  the  credit  of  not  only  stating  clearly  and  definitely  the 
principle  of  the  equivalence  of  work  and  heat,  but  also  of 
deducing  a  number  of  conclusions  from  it,  so  that  he  may  be 
regarded  as  the  founder  of  the  mechanical  theory  of  heat. 

He  claims  in  his  treatises  that  any  natural  force,  as  light 
or  heat,  cannot  be  destroyed,  either  in  whole  or  in  part,  any 
more  than  matter  itself.  That  what  appears  to  us  to  be 
destruction  is  nothing  more  than  a  transformation.  He  shows 
that  the  heat  received  from  the  sun  by  a  plant  enables  it  to 
extract  its  nourishment  from  the  air  and  earth — to  give  out 
oxygen  and  absorb  hydrogen  and  carbon — and  that  this  ab- 
sorbed hydrogen  and  carbon  can  furnish  again  the  same  quan- 
tity of  heat  received  by  the  plant.  That  just  as  these  elements 
furnished  heat  by  ordinary  combustion,  so  they  generated  heat 
in  animal  organisms,  and  enabled  them  to  perform  work,  to 
carry  loads,  etc.     He  points  out  that  the  carbon,  or,  in  other 


104  THERMO  DYNAMIC 8. 

words,  the  food,  which  an  animal  consnmes  is  proportional  to 
the  work  it  performs — that  an  increase  of  work  raises  the  tem- 
perature or  necessitates  an  increase  of  nourishment. 

Thus  he  computes  the  consumption  of  carbon  by  a  man  in 
climbing  a  mountain  10,000  feet  high,  and  puts  it  at  0.155  lbs. 
That  is,  the  heat  furnished  by  the  combustion  of  0.155  lbs.  of 
carbon  corresponds  to  the  mechanical  effect  of  raising  the  man 
10,000  feet.  Further,  he  determined  by  the  agitation  of  water 
in  a  vessel  the  work  necessary  to  raise  one  kilogram  of  water 
one  degree,  and  obtained  365  meter-kilograms,  which  is  thus 
considerably  less  than  the  result  obtained  by  Rumford. 

Exact  Determination  of  the  Meclianical  Equivalent  of  Heat  hy 
Joule  in  Mancliester. — ^Although  the  equivalence  between  work 
and  heat  was  clearly  expressed  and  proved  by  Mayer,  still 
there  was  wanting  an  exact  determination  of  the  amount  of 
work  necessary  in  order  to  heat  one  pound  or  one  kilogram  of 
water  one  degree.  Neither  the  result  of  Eumford  nor  that  of 
Mayer  can  lay  claim  to  great  accuracy.  Accordingly,  the  Eng- 
lish physicist.  Joule,  undertook  a  large  number  of  very  careful 
experiments,  in  order  to  determine  this  number  as  exactly  as 
possible.  He  adopted  different  methods.  He  caused  an  iron 
paddle-wheel,  set  in  motion  by  appropriate  cords  and  weights, 
to  revolve  in  a  vessel  filled  v/ith  water,  and  by  means  of  a  very 
accurate  thermometer  observed  the  increase  of  temperature 
of  the  water  when  the  weights  had  fallen  through  a  certain 
distance.  Then  he  took  otlier  liquids,  as  mercury,  oil,  etc., 
and  found  for  these  also  the  increase  of  temperature  for  a  cer- 
tain expenditure  of  work.  He  also  caused  two  cast-iron  plates, 
immersed  in  water,  to  rub  one  against  the  other,  and  compared 
here  also  the  rise  of  temperature  with  the  Avork  expended. 
In  another  series  of  experiments  he  forced  water  through  cap- 
illary tubes,  and  determined  the  rise  of  temperature  and  the 
work  expended.  Finally,  he  determined  the  increase  of  tem- 
perature when  air  is  compressed  in  a  receiver. 

From  all  these  experiments  he  found  that  by  the  expendi- 
ture of  a  certain  amount  of  work  an  equivalent  amount  of  heat 
was  generated,  entirely  independent  of  the  nature  of  the  sub- 
stances experimented  upon. 

After  seven  years  spent  in  study  of  the  subject,  he  under- 


MECHANICAL  EQUIVALENT.  105 

took,  in  1849,  another  large  series  of  experiments,  in  whicli  lie 
availed  himself  of  all  the  precautions  which  his  long  expe- 
rience had  made  familiar.  He  thus  found  for  the  work  neces- 
sary to  raise  one  pound  of  water  one  degree  Fahrenheit,  the 
following  results : 

772.692  foot  lbs.  for  friction  with  water — mean  of  40  experi- 
ments. 

774.083  foot  lbs.  for  friction  with  mercury — mean  of  50  ex- 
periments. 

774.987  foot  lbs.  for  friction  with  cast-iron — mean  of  20  ex- 
periments. 

When  we  consider  the  difficulties  attending  such  experi- 
ments, and  the  care  and  labor  required  for  the  exact  determina- 
tion of  the  work  and  temperature,  the  coincidence  of  these  re- 
sults is  most  remarkable.  These  experiments  belong,  in  fact, 
to  the  most  memorable  in  the  domain  of  physics,  and  entitle 
Joule  to  a  prominent  place  in  the  history  of  the  mechanical 
theory  of  heat. 

Of  all  these  resiilts.  Joule  considered  those  given  by  water 
as  the  most  reliable.  After  making  several  necessary  correc- 
tions, he  gave  the  equivalent  as  772  foot  lbs.  for  1"  Fahr.,  or 
772  X  -|  —  1389.6  foot  lbs.  necessary  to  raise  one  pound  of  water 
1°  Centigrade. 

Hence,  the  work  necessary  to  raise  one  kilogram  of  water 
one  degree  C.  is  1389.6  x  0.3048  ==■■  423.55  meter-kilograms. 

We  shall  hereafter  arrive  by  entirely  different  considera- 
tions at  almost  exactly  the  same  result,  so  that  we  may  regard 
it  as  settled  that  for  the  generation  of  one  heat  unit  *  a  work 
expenditure  of  423.55,  or,  in  round  numbers,  of  424  meter-kilo- 
grams is  necessary.  We  call  this  work  the  "  mechanical  equiv- 
alent OF  HEAT." 

Inversely,  by  the  expenditure  of  one  unit  of  work  we  can 
generate  only  i^^th  of  a  heat  unit,  and  this  number  we  may  call 
the  "  thermal  equivalent  of  ivorh" 

Since  one  horse-power  (French)  represents  75  meter-kilo- 
grams in  one  second,  we  have  -W  =  5.65  horse-power  necessary 
to  raise  one  kilogram  of  water  one  degree  in  one  second. 

*  The  term  "heatmiit"  is  iisutilly  used  to  denote  that  qnantity  of  heat  required  to  riiisc 
oneiMvnd  of  water  one  degree  Fahrenheit,  and  the  term  "  Calorie  "  is  employed  to  denote  that 
qnantity  of  heat  required  to  raise  one  kilogram  of  water  one  degree  Centigrade.  As  we  use 
throughout  this  work  French  units,  the  term  "  hcai  unit,"  unless  otherwise  stated  in  the  text, 
must  be  understood  as  meaning  the  French  heat  unit,  or  calorie. 


106  THEBMODTNAMIGS. 

EXAMPLE. 

Required  to  raise  2  kilograms  of  water  ic  20  minutes  from  0'  to  100°,  what 
expenditure  of  work  per  second  is  necessary  ? 

The  2  kilograms  require  100  x  2  =  200  heat  units,  or  a  work  of  424  x  200  = 
84,800  meter-kilograms.  Since  this  work  is  to  be  performed  in  20  minutes,  or 
1,200  seconds,  the  necessary  work  per  second  is  -/rinf  =  70.66  meter-kilograms 
per  second,  or  nearly  one  horse-power  (French).  (One  French  horse-power,  or 
75  meter-kilograms  per  second,  is  about  542.5  foot  lbs.  per  second,  or  somewhat 
less  than  an  English  horse-power,  which  is  550  foot  lbs.  per  second.) 

Hirn's  Determination  of  the  Mechanical  Equivalent. — Hirn,  a  civil 
engineer  at  Colmar,  lias  made  a  careful  and  difficult  determina- 
tion of  the  meclianical  equivalent  of  lieat.  He  adopted  a  differ- 
ent method  from  Joule,  and  attempted  to  determine  the  heat 
set  free  by  the  impact  of  inelastic  bodies. 

Two  heavy  blocks,  one  of  wood  the  other  of  iron,  were  sus- 
pended like  pendulums.  The  wooden  block  had  an  iron  plate 
upon  the  side  in  contact  with  the  other.  The  iron  block  was 
now  raised  a  certain  height,  and  a  hollow  lead  cylinder  placed 
upon  the  iron  plate,  which  was  struck  by  the  descending  iron 
block,  and  thus  compressed  and  heated. 

From  the  Aveight  of  the  iron  block,  and  the  height  through 
which  it  fell,  Hirn  calculated  its  living  force,  or  mechanical 
effect.  Moreover,  the  height  to  which  the  wood  block  was 
driven  by  the  shock,  as  well  as  that  to  which  the  iron  block 
rebounded,  gave  the  work  remaining  in  the  masses  after  impact. 
Now,  directly  after  the  shock,  the  lead  cylinder  was  filled  with 
water,  and  the  increase  of  temperature  of  the  water  determined. 
After  skillfully  determining  the  work  expended  in  the  compres- 
sion of  the  lead  cylinder,  it  was  easy  to  calculate  the  work  ex- 
pended in  the  heating.  The  mean  of  various  determinations 
gave  425  meter-kilograms,  or  the  same,  almost,  as  given  by  Joule. 

Performance  of  llechanical  Worh  hy  Heat. — In  the  preceding 
we  have  seen  that  heat  is  generated  by  work,  or,  generally,  that 
when  heat  appears  work  disappears,  and  that  by  a  certain  ex- 
penditure of  work  we  can  always  generate  a  certain  amount  of 
heat.  The  question  now  arises.  Can  we  generate  mechanical 
work  by  heat,  or  does  an  equivalent  amount  of  heat  always  dis- 
appear when  work  appears?  We  have,  at  once,  in  the  expan- 
sion of  solid  bodies,  a  most  striking  proof  of  the  performance 


WOBK  PERFORMED  BY  HEAT.  107 

of  work  by  heat.  Upon  the  surface  of  bodies  we  have  the  air 
pressure  of  about  10,334  kilograms  per  square  meter  (15  lbs. 
per  sq.  inch),  and  this  pressure  is  overcome  through  a  certain 
distance  during  the  expansion  of  the  body.  Since,  however, 
when  we  heat  a  body,  its  temperature  also  rises,  it  is  difficult 
to  determine  how  much  of  the  heat  imparted  goes  to  overcome 
this  pressure,  and  how  much  contributes  to  the  rise  of  tempera- 
ture. It  is,  therefore,  difficult  to  determine  from  the  expansion 
of  solid  bodies  the  relation  which  exists  between  the  heat  im- 
parted and  the  mechanical  work  obtained. 

The  gaseous  bodies  afford  the  easiest  proof  of  the  generation 
of  work  by  heat,  or  of  the  disappearance  of  heat  when  work  is 
obtained.  If,  for  example,  we  allow  compressed  air  to  issue 
from  a  receiver,  there  is  a  decrease  of  temperature,  sometimes 
so  great  that  drops  of  water  near  the  orifice  may  be  frozen. 
The  work  of  the  air  is  here  the  overcoming  of  the  outside  air 
pressure  as  it  expands. 

Joule  found,  by  similar  experiments,  a  work  performed  of  820 
foot  lbs.  for  every  degree  Fahr.  through  which  the  air  was 
cooled.  This  corresponds  to  |  x  820  x  0.3048  =  448.88  meter- 
kilogxams  for  every  degree  C,  or  somewhat  greater  than  already 
found.  The  method  of  determination,  however,  is  less  exact, 
and  the  discrepancy  was  to  be  expected. 

During  the  expansion  of  ordinary  atmospheric  air,  also,  heat 
disappears.  If  we  assume  air  of  atmospheric  pressure  under 
the  piston  EF  in  the  cylinder  AB  CD,  Fig.  8,  and      .  _ 

if  there  is  a  vacuum  above  it,  the  piston,  if  not 
held  fast,  will  rise,  while  the  temperature  of  the 
air  will  sink.  The  heat  which  thus  disappears 
is  the  equivalent  of  the  work  done  in  raising  the 
weight  of  the  piston  through  the  distance  trav- 
ersed. 

Although,  now,  heat  thus  disappears  when 
work  is  done  by  the  air,  as  when  it  rushes  into 
the  atmosphere  or  raises  a  weight,  we  should  not 
expect  to  find  any  such  disappearance  when  the 
air  expands  in  a  vacuum,  because  in  such  case  no 
work  is  performed.  This  point,  also,  Joule  has  experimen- 
tally investigated.* 

*  See,  also,  page  22  of  Introduction. 


108  THEBMODTNAMICS. 

He  made  use  of  two  copper  vessels  connected  by  a  pipe. 
Botli  were  placed  in  .a  vessel  of  water,  the  temperature  of  which 
was  determined  directly  before  and  after  the  experiment.  Con- 
nection being  closed,  the  air  was  compressed  to  22  atmospheres 
in  one  vessel,  and  was  exhausted  from  the  other.  The  cock 
was  then  turned,  and  the  cojnpressed  air  rushed  into  the  empty 
vessel  until  the  pressure  was  the  same  in  both.  There  was 
found  to  be  neither  a  rise  nor  fall  of  temperature  while  the 
transfer  took  place,  the  temperature  of  the  water  in  which  the 
vessels  were  immersed  remaining  the  same  both  before  and 
after  the  experiment. 

We  may  explain  this  as  follows :  As  soon  as  a  part  of  the 
air  has  passed  from  one  vessel  into  the  other,  the  air  continu- 
ing to  enter  must,  to  be  sure,  perform  work  in  compressing  the 
air  already  there.  There  must  be,  therefore,  in  the  vessel  from 
which  the  air  flows,  a  decrease  of  temperature,  but  in  the  other, 
where  the  air  is  being  compressed,  an  increase  of  temperature 
precisely  equal,  so  that,  on  the  whole,  heat  is  neither  lost  nor 
gained. 

That  this  is  actually  the  true  explanation  was  proved  by 
inclosing  the  vessels  in  two  separate  cisterns,  and  observing 
the  temperature  of  each,  both  before  and  after  the  experiment. 

It  is  therefore  proved,  at  least  for  air,  that  when  heat  dis- 
appears work  is  gained ;  there  is  also  no  room  for  doubt  that 
for  every  unit  of  heat  disappearing,  the  same  mechanical  work 
is  gained  which  we  have  already  found  to  be  necessary  for  the 
generation  of  one  unit  of  heat,  even  although  Joule  found  for 
each  disappearing  heat  unit  a  somewhat  greater  work.  One 
heat  unit  is  therefore  equivalent  to  a  mechanical  work  of  424 
meter-kilograms. 

But  Hirn  has  proved  the  correctness  of  this  princijDle  for 
steam  also,  by  means  of  the  steam  engine.  In  fact,  this  machine 
shows  us  at  once  how  heat  can  be  transformed  into  work,  and 
it  was  this  especially  which  early  led  physicists  to  the  conclu- 
sion that  there  must  be  a  certain  equivalence  between  heat  and 
mechanical  work. 

Him  determined  first  the  temperature  and  tension  of  the 
steam  in  the  boiler  and  the  steam  consumption  per  stroke.  He 
was  thus  able  to  determine  the  number  of  heat  units  carried 
per  stroke  to  the  cylinder.    Then  he  observed  the  temperature 


WORK  PERFORMED  BY  SEAT.  109 

and  quantity  of  tlie  condensing  water  used  for  each  stroke,  as 
well  as  tlie  temperature  of  tliis  water  after  tlie  condensation 
of  the  steam.  He  was  tlius  able  to  determine  tlie  loss  of  heat 
experienced  by  the  steam  during  its  action  in  the  cylinder,  as 
well  as  that  due  to  the  resistances  in  the  steam  pipe,  in  the 
Yalve  box,  and  in  the  exhaust  pipe.  Finally,  he  endeavored  to 
estimate  the  loss  of  heat  due  to  conduction  and  radiation  be- 
tween the  boiler  and  the  condenser.  He  then  determined  the 
mechanical  work  imparted  to  the  fly-wheel,  and  also,  as  accu- 
rately as  possible,  the  work  absorbed  by  the  prejudicial  resist- 
ances, such  as  the  friction  of  the  piston,  the  slide  valve,  etc. 
He  Avas  thus  able  to  calculate  the  work  for  each  unit  of  heat 
disappearing  in  the  cylinder.  He  found  as  a  mean  of  many 
determinations  413  meter-kilograms,  a  number  which  agrees 
quite  closely  with  Joule's  results,  especially  when  we  con- 
sider the  very  great  difficulties  which  Hirn  had  to  contend 
with. 

Still  another  aj^paratus  affi^rds  us  clear  proof  that  heat  can 
be  transformed  into  mechanical  work,  viz.,  the  Gifford  injector 
—  an  apparatus  which  has  for  several  years  been  used  for 
furnishing  feed  water  to  locomotives  and  stationary  engines. 
In  this  contrivance  a  pipe  leads  from  the  steam  space  in  the 
boiler  to  the  water  space.  This  pipe  joins  another  at  a  suitable 
place,  which  brings  the  feed  water.  The  steam  flows  toward 
the  water  space  and  forces  the  air  with  it,  A  partial  vacuum 
is  thus  caused  in  the  feed-water  pipe,  and  the  water  rises  in  it 
until  it  meets  the  steam  current.  By  this  it  is  carried  along 
with  great  velocity  and  forced  into  the  boiler. 

Since,  now,  the  water  in  the  boiler  is  acted  upon  by  the 
steam  pressure,  it  is  not  at  once  evident  how  it  can  be  possible 
that  the  steam  flows  round  toward  the  water,  instead  of  the 
water  being  forced  into  the  steam  space.  When  we  remember, 
however,  that  the  steam  contains  a  large  amount  of  latent  heat, 
which  the  water  does  not  possess,  or,  in  other  words,  that  the 
work  inherent  in  the  steam  is  greater  than  that  in  the  water, 
we  can  easily  understand  the  possibility  of  the  action. 

When  the  steam  comes  in  contact  with  the  cold  feed  water, 
a  part  of  the  latent  heat  goes  to  heat  the  water,  but  another 
part  is  transformed  into  mechanical  work,  and  this  it  is  which 
forces  the  water  into  the  boiler. 


no  THERMODYNAMICS. 


QUESTIONS  FOE  EXAJVIINATION. 

Give  instances  of  heat  generated  by  mechanical  action.  What  conclusion  do  such  instances 
point  to  ?  Describe  experiments  which  test  this  conclusion.  What  did  Count  Rumford  con- 
clude ?  What  chances  for  error  were  there  in  his  experiment  ?  Describe  Davy's  experiment. 
What  precautions  did  he  take  against  possible  objections  ?  Do  you  regard  his  experiment  as 
conclusive  ?  W^hy  ?  What  was  the  part  played  by  Mayer  as  regards  the  theory  ?  In  what 
respect  was  it  diHerent  from  that  of  Rumford  and  Davy  ?  What  conclusions  did  he  deduce  from 
his  views  ?  Who  first  made  an  exact  determination  of  the  mechanical  equivalent '?  Describe 
some  of  the  methods  he  employed.  How  long  did  he  devote  himself  to  the  subject  ?  What  did 
he  consider  on  the  whole  the  most  exact  result  ?  Define  precisely  what  you  understand  by  the 
"mechanical  equivalent  of  heat."  Illustrate.  What  is  a  " foot-pound  ?  "  What  is  a  "  horse- 
power ?  "  What  aie  the  equivalent  French  units  of  measurement  ?  What  is  a  meter-kilogram  ? 
What  constitutes  a  French  horse-power  ?  How  does  it  differ  from  the  English  ?  What  system 
is  used  in  this  work  ?  Define  exactly  what  you  understand  by  a  "heat  unit."  What  do  you 
understand  by  "  calorie  ?  "  Do  we  use  the  term  calorie  in  this  book  ?  AVhy  not  ?  What  is  the 
mechanical  equivalent  in  foot-pounds  for  Fahrenheit  scale  ?  What  for  Centigrade  scale  ?  What 
is  it  in  French  measure  ?  Show  how  to  reduce  one  to  the  other.  Define  exactly  what  you 
understand  by  "  thermal  equivalent  of  work."    Illustrate. 

What  work  is  necessary  to  raise  2  kilograms  of  water  from  0°  to  100°  C.  ?  What  horse-power 
(French)  is  required  to  perform  this  work  in  20  minutes  ?  What  work  is  necessary  to  raise  2 
pounds  of  water  from  0°  to  100°  C.  ?  From  0°  to  100°  F.  ?  What  horse-power  is  required  in  each 
case  to  perform  these  works  in  20  minutes  ?  How  do  you  reduce  Centigrade  degrees  to  Fahren- 
heit, and  vice  versa?  Define  "work."  What  other  term  is  sometimes  employed?  What 
methods  did  Hirn  employ  in  order  to  determine  the  mechanical  equivalent  ?  Did  it  confirm  the 
result  obtained  by  Joule  ?  Is  the  mechanical  equivalent  constant  ?  Were  it  to  vary,  what  impos- 
sible result  could  you  logically  deduce  ?  (See  latrocluctinn.)  Deduce  this  consequence.  {Intr.) 
Is  perpetual  motion  possible  ?  Why  not  ?  (Intr.)  Are  you  fully  satisfied  of  this  impossibility? 
(Intr.,  Notes.)    Can  any  scheme  for  attaining  it  rightfully  claim  attention  ?    Why  not  ? 

Give  instances  of  mechanical  work  performed  by  heat.  Describe  Joule's  experiments  in  this 
direction.  Wlien  air  expands  why  does  its  temperature  fall  ?  When  it  expands  into  a  vacuum 
does  its  temperature  fall  ?  Why  not  ?  Describe  Joule's  experiment  here.  Under  what  circum- 
stances does  the  temperature  remain  constant  in  this  experiment  ?  Under  what  does  it  vary  ?  Is 
this  in  full  accord  with  our  principle  ? 

Who  first  proved  this  principle  for  steam  ?  How  was  it  done  ?  What  sources  of  error  must 
be  guarded  against  ?  Enumerate  in  order  the  principal  steps  of  the  experiment.  Does  this 
experiment  afford  additional  proof  of  the  correctness  of  our  principle  ?  What  other  apparatus 
illustrates  it  ?  Explain  the  secret  of  its  action.  Are  you  convinced  that  our  principle  is  the 
expression  of  a  natttral  law  ?  What  do  you  understand  b3' a  "  natural  law?"  State  concisely 
the  law  of  ■  equivalence  of  heat  and  work.  What  do  j'ou  understand  bj' "  equivalence  ?  "  Are 
things  equivalent  necessarily  identical  ?  Is  otir  numerical  determination  exact  ?  If  not,  would 
that  affect  your  belief  in  the  truth  of  the  law  ?  Why  ?  What  did  Davy  and  Rumford  suspect 
"  heat "  to  consist  of  ?  What  do  you  understand  by  "  work  ?  "  If  their  suspicions  were  correct, 
how  would  "heat"  differ  from  "work  ?"  Does  the  proof  or  truth  of  our  principle  rest  upon 
any  such  hypothesis  ?  Does  it  rest  upon  any  hypothesis  ?  Are  you  fli-mly  convinced  of  its  gener- 
ality and  truth  ? 


CHAPTEE  II. 

HEAT  A  KIND   OF  MOTION. 

Feom  what  has  been  said,  we  may  regard  it  as  settled  that 
with  a  certain  amount  of  mechanical  work  we  can  always  gen- 
erate a  certain  amount  of  heat,  and  inversely,  that  with  a  cer- 
tain amount  of  heat  a  definite  amount  of  mechanical  work  can 
be  produced ;  that  the  amount  of  heat  which  can  raise  the  tem- 
perature of  one  kilogram  of  water  one  degree  is  equivalent  to 
a  mechanical  work  of  424  meter-kilograms,  and  that,  inversely, 
by  the  expenditure  of  this  work  one  kilogram  of  water  can  be 
heated  one  degree. 

If,  now,  such  an  equivalence  exists  between  heat  and  mechan- 
ical work,  we  are  com23elled  to  assume  that  heat  is  properly 
nothing  else  than  mechanical  work  in  another  form.  And  since 
heat  is  possessed  by  bodies,  since  it  can  enter  from  without 
into  bodies,  we  must  expect  to  find  here  something  similar  or 
equivalent  to  mechanical  work.  We  must  therefore  assume 
that  either  the  atoms  of  a  body,  or  the  molecules  or  groups  of 
atoms,  or  that  some  other  substance  between  the  atoms  is  set 
into  some  sort  of  motion.  Such  a  substance  is  the  ether,  which 
is  universally  regarded  as  the  medium  for  the  transmission  of 
light. 

One  theory,  applicable  to  gaseous  bodies,  was  first  pro- 
pounded by  Daniel  Bernoulli.  This  was  further  developed  by 
Kronig  and  Clausius,  and  applied  to  solid  and  liquid  bodies 
also.     It  numbers  at  present  the  most  adherents. 

Another  theory,  which  was  at  an  earlier  period  the  most 
widely  accepted,  is  due  especially  to  Poisson  and  Cauchy,  and 
was  developed  principally  by  Redtenbacher. 

Bedtenbaclier  s  Theory. — According  to  Redtenbacher  the  body 
atoms  attract  each  other,  while  between  the  ether  atoms  there 
is  a  mutual  repulsion.      The   body  atoms  are  incomparably 

111 


112  THERMODYNAMICS. 

larger  than  tlie  ether  atoms,  and  these  last  have  inertia,  but 
are  not  affected  by  gravity. 

In  consequence  of  these  properties  each  body  atom  is  sur- 
rounded with  an  envelope  of  ether  atoms.  In  proximity  to  the 
atom  this  envelope  has  its  greatest  density,  which  diminishes 
with  the  distance  from  the  center.  Thus  the  ether  surrounds 
each  atom  just  as  the  atmosphere  surrounds  the  earth,  pro- 
vided that  the  atoms  are  round.  An  atom  with  its  envelope 
Kedtenbacher  calls  a  "  dynamide."  Between  every  two  dyna- 
mides  there  is  a  space  incomparably  greater  than  the  dynamide. 

The  heat  of  a  body  consists  now  in  a  pulsating  motion  of  the 
ether  envelopes.  It  is  assumed  that  each  ether  atom  vibrates 
in  the  direction  of  the  radius  of  the  dynamide.  The  quicker 
and  stronger  the  pulsations  of  the  envelope,  so  much  the  hotter 
is  the  body. 

If  the  attraction  of  the  body  atoms  is  equal  to  the  repulsion 
of  the  ether  envelopes,  the  body  is  either  solid  or  liquid.  If, 
on  the  other  hand,  the  repulsive  force  of  the  latter  is  greater 
than  the  attractive  force  of  the  former,  the  body  is  gaseous. 

In  the  solid  state  the  body  atoms  are  generally  closer  to- 
gether than  in  the  liquid  ;  their  mutual  attractions  are  greater, 
the  ether  envelopes  are  denser,  and  their  atoms  vibrate  through' 
shorter  distances.  Displacement  of  the  particles  is  thus  dif- 
ficult. A  complete  union  of  the  atoms,  however,  cannot  occur, 
for  the  nearer  the  atoms  approach,  the  greater  is  the  repulsion 
of  the  ether  envelopes,  until  finally  there  is  again  equilibrium 
between  attraction  and  repulsion. 

As  now  the  ether  envelopes  of  a  solid  body  pulsate  faster 
and  stronger,  or  as  the  body  becomes  warmer,  the  body  atoms 
recede  from  each  other,  their  attraction  becomes  less,  until 
finally  ever  so  small  a  force  can  displace  the  particles.  The 
body  is  then  liquid. 

When  the  ether  envelopes  no  longer  pulsate,  that  is,  when 
there  is  no  motion  within  the  body,  the  body  possesses  no 
heat — it  is  then  absolutely  cold.  We  shall  soon  see  to  what 
degree  of  the  thermometer  this  "  absolute  zero  "  of  temperature 
corresponds. 

Other  Vieios  as  to  the  Nature  of  Heat. — The  other  view  as  to  the 
nature  of  heat  starts  also  with  the  assumption  that  the  atoms, 


HEAT  A  KIND   OF  MOTION.  113 

or  groups  of  atoms,  tlie  so-called  molecules,  are  set  into  vibra- 
tory motion.  This  view,  as  already  indicated,  was  indorsed  by 
the  famous  English  chemist  Sir  Humphry  Davy.  He  says,  in 
his  treatise  upon  heat  and  light,  that  heat,  or  that  force  which 
prevents  the  direct  contact  of  the  body  atoms,  and  gives  rise  to 
the  sensations  of  heat  and  cold,  may  be  defined  as  a  special, 
and,  in  all  probability,  vibrating  motion  of  the  body  atoms 
which  tends  to  separate  them.  This  may  be  called  a  repulsive 
action.  Since  there  is  also  an  attractive  force,  we  may  con- 
ceive the  body  atoms  as  acted  upon  by  two  opposite  forces, 
that  of  attraction  and  that  of  repulsion.  The  first  of  these 
forces  is  the  combined  action  of  cohesion,  which  strives  to  hold 
the  atoms  in  contact ;  of  gravity,  which  tends  to  collect  them 
into  masses,  and  of  the  pressure  exerted  by  exterior  bodies. 
The  second  of  these  forces  is  due  to  a  certain  vibrating  motion 
which  tends  to  keep  them  apart,  and  which  may  be  generated, 
or  rather  increased,  by  friction  or  impact.  The  action  of  cohe- 
sion, in  causing  the  body  atoms  to  approach,  is  precisely  similar 
to  the  attraction  of  gravitation  upon  the  large  masses  of  the 
universe,  and  the  repulsive  force,  to  the  centrifugal  force  of  the 
planets.  In  his  "Chemical  Philosophy"  also,  Davy  says  that 
all  the  phenomena  of  heat  can  be  explained  by  assuming  that 
in  solid  bodies  the  atoms  are  in  a  permanent  condition  of  vibra- 
tory motion,  and  that  these  vibrations  become  more  rapid  and 
larger  as  the  temperature  increases.  In  liquid  and  gaseous 
bodies  we  must  assume  that  the  atoms,  besides  their  vibratory 
motion,  have  also  a  motion  around  their  axes,  and  that  both 
these  motions  are  greatest  in  gaseous  bodies. 

According  to  Clausius,  who  has  contributed  much  to  the 
development  of  the  mechanical  theory  of  heat,  it  is  less  the 
atoms  themselves  than  the  groups  of  atoms,  or  the  molecules, 
which  are  in  motion.  The  manner  in  which  the  atoms  combine 
to  form  a  molecule,  and  the  form  of  the  same,  determine  the 
properties  of  the  body.  Hence  it  may  happen  that  the  same 
simple  bodies  or  elements  possess  now  these,  now  those  prop- 
erties, and  thus  we  may  have  the  so-called  allotropic  condi- 
tion. It  is  evident  that  in  chemically  compound  bodies  also 
there  may  be  such  a  various  grouping  of  atoms ;  indeed,  in  such 
case  the  variety  of  grouping  can  be  even  greater. 

While  now,  in  solid  bodies,  there  is  in  all  probability  a  vibrat- 


114  THERMODTNAMICS. 

ing  motion  of  the  molecules,  we  are  obliged  to  assume  in  liquids 
a  rotating  motion  also,  since  in  sucli  bodies  a  molecule  easily 
separates  from  its  neighbors. 

The  vaporization  of  liquids,  according  to  this  view,  may  be 
explained  as  follows : 

As  heat  is  imparted  to  a  liquid  so  that  the  velocity  of  vibra- 
tion becomes  greater  and  greater,  and  the  force  of  cohesion 
ever  less  and  less,  the  molecules  finally  break  loose  and  move, 
like  the  particles  of  gaseous  bodies,  in  straight  lines.  This 
may  be  illustrated  by  an  experiment.  Suppose  at  the  end  of  a 
spiral  spring  a  weight  fastened,  and  one  end  is  held  in  the  hand 
while  the  weight  is  swung  round  in  a  circle.  The  spring  will 
be  extended  and  the  weight  will  recede  as  the  velocity  increases. 
Finally  the  centrifugal  force  becomes  greater  than  the  tenacity 
of  the  spring,  which  then  breaks,  and  the  weight  flies  off  in  a 
tangential  direction. 

As  the  attractive  force  of  the  molecules  is  not  the  same  for 
different  liquids,  some  require  more,  some  less  heat  to  convert 
them  into  vapor. 

It  is  also  evident  that  vaporization  takes  place  more  rapidly 
at  the  surface  of  a  body  than  in  the  interior,  because  the  mole- 
cules are  not  restrained  by  the  pressure  of  those  above.  Thus, 
if  the  centrifugal  force  of  a  molecule  in  the  interior  of  a  body  is 
not  sufficient  to  overcome  the  action  of  those  above  it,  it  is 
obliged  to  retain  its  earlier  vibratory  or  rotating  motion  which 
it  possessed  as  a  liquid  molecule.  In  such  case  we  say  that 
the  liquid  "  simmers."  Further,  it  is  evident  that  the  air  mole- 
cules which  impinge  against  the  surface  of  the  liquid  must 
partially  impede  the  process  of  vaporization.  We  say  partially, 
because  we  assume  that  there  are  large  spaces  between  the  air 
molecules  through  which  the  liquid  molecules  can  move.  Yet 
it  may  often  happen  that  an  air  molecule  comes  in  contact  with 
a  liquid  molecule  just  as  it  is  about  to  leave  the  surface.  If 
therefore  the  density  of  the  air  diminishes,  or  if  it  is  entirely 
removed,  the  vaporization  goes  on  quicker  for  the  same  tem- 
perature, more  particles  breaking  loose.  These  results  are,  as 
we  know,  confirmed  by  experience. 

When  vaporization  takes  place  in  the  interior  of  a  liquid,  the 
centrifugal  force  of  the  molecules  must  be  so  great  as  to  over- 
come the  pressure  upon  them.     This  pressure  consists  not  only 


HEAT  A  KIND   OF  MOTION.  115 

of  tlie  weight  of  superincumbent  liquid,  but  also  of  the  press- 
ure of  the  air  upon  the  surface  of  the  liquid.  The  centrifugal 
force  must  therefore  be  greater  the  deeper  the  molecules  lie 
beneath  the  surface.  If  they  still  break  loose,  their  centrifugal 
force  must  be  equal  to  the  pressure  of  the  air,  increased  by  the 
weight  of  superincumbent  liquid — and  hence  it  may  be  that, 
for  instance,  in  deep  vessels  the  same  mass  of  water  is  made  to 
boil  with  more  difficulty  than  in  shallow. 

Let  us  now  notice  briefly  how  combustion  is  regarded.  We 
may  call  attention,  first,  to  the  following  observations : 

If  we  allow  a  weight  to  fall  from  a  certain  height  upon  a 
plate  of  lead  or  iron,  its  living  force  is  entirely  or  joartially 
destroyed  by  the  impact,  according  as  the  bodies  are  entirely 
or  partially  inelastic.  But  if  we  now  examine  the  plate,  we 
shall  find  that  it  has  been  heated,  and  heated  more  according 
as  more  of  the  living  force  of  the  weight  has  disappeared.  The 
mechanical  work  inherent  in  the  weight  has  thus  been  trans- 
formed after  impact  into  heat. 

In  combustion  the  process  is  similar,  but  instead  of  large 
m.isses  we  have  here  to  do  with  body  particles  which  elude 
observation  even  with  the  microscope.  Thus  when  we  heat, 
for  example,  pure  carbon,  the  particles  are  not  only  set  into 
much  more  rapid  vibration,  that  is,  their  living  force  increased, 
but,  at  the  same  time,  they  recede  from  each  other.  It  is  thus 
more  easy  for  the  impinging  oxygen  atoms  to  penetrate  between 
the  atoms  of  carbon,  and  by  reason  of  greater  adhesion  to  unite 
with  them.  By  this  impact  the  rectilinear  motion  of  the  oxy- 
gen atoms  is  transformed  into  the  vibratory  or  rotating  motion 
of  the  carbon  molecules,  that  is,  generates  heat.  At  the  same 
time  every  carbon  atom  unites  with  two  oxygen  atoms  to  make 
a  molecule  of  carbonic  acid,  which  then  takes  a  rectilinear 
motion. 

Heat  Conduction  and  Radiation. — In  the  foregoing  we  have 
briefly  reviewed  the  dijBferent  views  as  to  the  nature  of  heat, 
and  have  sought  to  explain  some  of  the  best  known  phenomena 
in  accordance  with  these  views.  Let  us  now  inspect  somewhat 
more  closely  the  phenomena  of  radiation  and  conduction,  in 
order  to  deduce  that  the  heating  of  a  body  is  not  so  much  the 
motion  of  ether  envelopes,  hut  rather  consists  in  a  greater  velocity  of 


116  THERMODYNAMICS. 

vibration  of  the  body  molecules.  Althougli  the  facts  are  familiar 
to  us  all,  yet  it  may  not  be  uninteresting  to  call  attention  to 
tliem  again. 

We  say  that  a  body  is  heated  by  conduction  when  it  receives 
heat  from  another  with  which  it  is  either  in  direct  contact  or 
by  the  intervention  of  another. 

In  such  case  the  molecules  of  the  warmer  body  impart  their 
greater  living  force  in  part  to  those  of  the  colder,  until  the 
molecules  of  both  possess  the  same  living  force,  or  are  equally 
warm. 

A  body  is  heated  by  radiation  when  it  receives  heat  from 
another  without  the  intervention  of  a  third.  If,  for  example, 
we  stand  near  a  hot  stove,  or  expose  ourselves  to  the  sun's  rays, 
we  receive  heat  even  when  the  surrounding  air  is  cold.  Heat 
rays  must  therefore  proceed  from  the  source  of  heat,  just  like 
light  rays  from  a  source  of  light,  which  rays  excite  in  us  the 
sensation  of  heat. 

The  researches  of  physicists  have  shown  that  these  heat  rays 
follow  the  same  laws  as  those  of  light ;  that,  for  example,  they 
are  in  similar  manner  reflected  and  refracted,  and  show  the 
same  phenomena  of  interference  which  have  made  us  acquainted 
with  the  nature  of  light. 

We  may  at  present  assume  with  certainty  that  light  rays  are 
propagated  in  a  similar  manner,  and  arise  in  a  similar  manner, 
to  sound  waves  in  the  air  or  water  waves  in  the  water. 

Luminous  bodies  possess  the  power  of  putting  into  vibration 
the  ether  which  pervades  all  space  and  all  bodies.  These  vibra- 
tions reach  our  eyes,  excite  the  retina,  and  thus  cause  sight. 
Just  as  high  tones  are  caused  by  quicker,  and  low  tones  by 
slower  vibrations  of  the  sounding  body  and  of  the  air,  so  differ- 
ent colors  are  caused  by  quicker  and  slower  vibrations  of  the 
ether. 

Since,  now,  radiant  heat  follows  the  same  laws  as  light,  tve 
must  attribute  its  origin  and  transmission  to  the  vibrations  of  the 
ether  atoms.    Of  this  the  following  experiment  will  convince  us  : 

If  we  allow  a  ray  of  light  to  enter  a  dark  room  through  an 
aperture,  and  to  pass  through  a  triangular  prism  held  in  front 
of  the  aperture,  the  ray  will  be  not  only  deviated  or  refracted, 
but  the  ray  originally  white  will  be  split  up  into  various  colors, 
of  which  we  may  distinguish  especially  seven,  the  so-called 


HEAT  A   KIND   OF  MOTION.  117 

prismatic  colors,  or  colors  of  tlie  rainbow.  All  the  colors  bear 
the  name  of  "  spectrum,"  and  when  obtained  by  the  decomposi- 
tion of  the  sun's  rays,  of  the  solar  spectrum. 

Without  the  prism,  we  obtain  a  light  strip  upon  the  screen, 
of  the  form  of  the  aperture,  in  the  straight  line  drawn  from  the 
aperture  to  the  sun.  With  the  prism,  the  light  strip  is  very 
much  broader,  is  deviated  toward  the  refracting  angle  of  the 
prism,  and  is  vividly  colored.  The  colors,  beginning  with  those 
least  refracted,  are  red,  orange,  yellow,  green,  blue,  indigo  blue, 
and  violet. 

If  we  examine  the  spectrum  formed  upon  the  screen  with  a 
sensitive  thermometer,  or  a  thermo-electric  pile,  w^hich  shows 
least  diflferences  of  temperature,  we  shall  find  that  the  violet 
rays  have  the  least  heating  power,  and  the  red  the  greatest. 
Indeed,  if  we  examine  beyond  the  red  raj^s,  where  no  color  can 
be  perceived,  we  find  that  the  heat  is  still  greater  than  in  the 
siDectrum  itself,  and  these  invisible  rays  extend  beyond  the 
spectrum  a  distance  about  equal  to  its  own  length.  Thus  the 
so-called  heat  spectrum  has  about  double  the  length  of  the 
color  spectrum. 

Hence  it  appears  that  the  heat  rays  are  less  refracted  than 
the  light  rays,  since  they  are  less  deviated. 

For  the  sake  of  completeness  we  may  also  add  the  following : 

If  we  extend  our  examination  beyond  the  violet,  we  find  here 
rays  which  have  the  greatest  influence  ujoon  chemical  combina- 
tions. In  fact,  these  rays  have  the  most  powerful  photographic 
effect,  and  hence  they  constitute  the  chemical  spectrum. 

Physicists  have  computed  that  for  red  light  the  ether  must 
make  not  less  than  481  billion  vibrations  in  a  second,  while  for 
violet  light  764  billions  are  necessary.  The  heat  rays,  then, 
are  caused  by  ether  vibrations,  on  the  whole  less  than  481  bil- 
lion per  second.  On  the  other  hand,  the  number  of  vibrations 
of  the  chemical  rays  is  more  than  764  billion  per  second. 

When,  now,  the  vibrating  ether  meets  a  body,  it  sets  in  vi- 
bration either  the  ether  existing  in  the  body,  or  the  atoms  of 
the  body,  or  both.  If  the  ether  in  a  body  is  so  constituted  that 
it  can  transmit  the  vibrations  which  it  receives  from  white  light, 
it  is  transparent  and  colorless.  If  it  can  only  transmit  those 
vibrations  which  it  receives  from  the  violet  rays,  while  the 
others  are  lost,  it  appears  violet  in  color,  and  so  on.     In  an 


118  THEBMOD  YNA  MIG8. 

opaque  body  tlie  ether  is  so  constituted  that  it  cannot  trans- 
mit the  vibrations  communicated  to  it  by  a  source  of  light. 

While,  now,  those  ether  vibrations  generated  by  a  source  of 
light,  and  which,  we  must  assume,  possess  a  greater  velocity 
but  a  less  amplitude  of  vibration  than  heat  rays,  set  into  vi- 
bration the  ether  contained  in  bodies,  it  is  those  vibrations 
which  arise  from  a  source  of  heat,  and  which  have  a  less  veloc- 
ity but  a  greater  amplitude  of  vibration,  which,  chiefly  by  im- 
pact, set  the  atoms  of  a  body  into  vibration,  and  thus  heat  it. 
Sinc9,  however,  the  heat  spectrum  is  about  double  the  length 
of  the  light  spectrum,  and  since,  therefore,  there  are  heat  rays 
of  different  refrangibility  and  velocity  of  vibration,  we  might 
expect  that  the  molecules  of  one  and  the  same  body  are  differ- 
ently excited  by  different  heat  rays,  and  the  body  differently 
heated — indeed,  that  certain  heat  rays  may  not  excite  the  mole- 
cules of  certain  bodies  at  all,  but  only  the  contained  ether,  and 
thus  that  these  bodies  are  not  heated  by  such  rays.  Science 
furnishes  a  number  of  confirmations  of  this  conclusion. 

If,  for  example,  we  let  fall  upon  a  plate  of  fluor-spar  of  about 
2.6  millimetei's  in  thickness,  rays  from  various  sources,  such  as 
the  electric  lamp,  glowing  platinum,  heated  copper,  the  heat 
transmitted  is  found  to  be  in  each  case  as  the  numbers 

78  :  69  :  42. 

Thus  the  rays  from  the  lamp  excite  the  ether  in  the  fluor- 
spar more  than  the  atoms  or  molecules,  while  those  from  the 
copper  act  inversely. 

Still  more  striking  is  the  transmission  of  different  heat  rays 
by  beryl,  calkspar,  rock-crystal,  etc. 

Different  bodies  also  transmit  different  amounts  of  heat  fi'om 
one  and  the  same  source.  Thus,  for  example,  clear  rock-salt 
transmits  92  per  cent,  of  the  rays  from  the  electric  lamp,  fluor- 
spar, on  the  other  hand,  only  78,  colorless  alum  9,  and  very 
clear  ice  only  6  per  cent. 

Qf  all  bodies  rock-salt,  and  air  especially,  transmit  equally 
well  rays  from  different  sources. 

Following  Melloni,  an  Italian  physicist  to  whom  we  are  in- 
debted for  valuable  researches  upon  radiant  heat,  we  call  those 
bodies  which  transmit  the  heat  rays,  diathermanous,  and  those 
which  do  not,  athermanous. 


HEAT  A  KIND  OF  MOTION.  119 

If  a  body  cannot  transmit  tlirougli  its  mass  the  heat  rays 
which  strike  its  surface,  these  rays  are  either  reflected,  accord- 
ing to  the  same  laws  as  light  rays,  or  they  are  absorbed  and 
heat  the  body.  The  greater  the  amount  of  heat  transmitted, 
the  less  is  that  absorbed.  Thus,  for  example,  rock-salt  becomes 
heated  but  little  when  exposed  to  the  sun's  rays  or  other  sources 
of  heat,  because  it  is  very  diathermanous,  while  smoked  glass, 
which  transmits  but  few  rays,  is  heated  much  more. 

Hence  it  follows  that  in  absorption  of  heat,  also,  there  is  con- 
siderable diflerence  among  bodies,  and  that  the  same  body  does 
not  receive  equal  amounts  of  heat  from  different  sources.  Thus 
while,  for  example,  soot  absorbs  almost  completely  the  rays 
from  the  electric  light,  white  lead  absorbs  only  0.53,  shel-lac 
only  0.43,  and  a  metal  surface  only  0.14 

On  the  other  hand,  the  absorption  power  of  white-lead  for 
rays  from  the  electric  light,  from  glowing  platinum,  or  from 
copper  heated  up  to  400°,  is  as 

53  :  56  :  89. 

The  same  holds  for  most  other  bodies.  Soot  alone  seems  to 
absorb  equal  amounts  of  heat  from  all  sources.  Just  as  in 
respect  to  light  it  is  perfectly  opaque,  so  is  it  in  respect  to  the 
heat  rays. 

If,  now,  we  investigate  the  heat  rays  which  a  body  heated  by 
absorption  again  emits,  we  shall  find  that  these  rays  are  en- 
tirely independent  of  the  nature  or  quality  of  those  absorbed. 
Yery  extensive  experiments  have  shown  that  the  heat  rays 
emitted  by  a  body  are  alicays  the  same,  ivhatever  the  source  from 
which  the  body  is  heated,  lohether  the  electric  lamp,  gloiuing  platinum, 
or  even  contact  loith  a  toarmer  body. 

This  seems  to  indicate  plainly  that  the  heat  rays  absorbed 
by  a  body  are  completely  altered  in  character.  It  seems  also 
indicated  by  the  fact  that  the  absorbed  heat  is  propagated  very 
slowly  in  the  interior  of  a  hodj,  while  the  radiant  heat  has  a 
velocity  equal  to  that  with  which  heat  travels  through  air  and 
space — a  velocity  probably  not  less  than  that  of  light  itself. 

We  can  therefore  hardly  assume  that  the  heating  of  a  body 
by  absorption  of  heat  rays  consists  in  setting  the  body  ether 
into  more  rapid  vibration ;  for  in  such  case  it  would  be  hard  to 
see  why  the  heat  is  not,  as  in  diathermanous  bodies,  entirely  or 


120  THEBMOBYNAMlCa. 

partially  transmitted,  and  wliy  tlie  propagation  in  tlie  body  is 
so  slow.  It  is  mucli  more  probable  that  the  atoms  or  molecules 
of  athermanous  bodies  are  excited  by  the  impinging  heat  rays,  and 
put  into  quiclcer  vibration,  and  that  this  is  the  cause  of  the  heating. 
This  conclusion  seems  not  without  interest. 


QUESTIONS  FOR  EXAMINATION. 

What  amount  of  work  is  required  to  raise  the  temperature  of  one  kilogram  of  water  one 
degree  ?  How  many  pounds  make  a  kilogram  ?  Wh;U  two  theories  are  there  ?  Give  Redten- 
baclier's  theory'.  What  do  you  understand  by  "  atom  ?  "  What  by  "ether  ?  "  What  by  "  mole- 
cule ?  "  What  by  "  dynamide  ?  "  In  what  does  the  heat  of  a  body  consist  according  to  Redten- 
bacher  ?  Under  what  conditions  is  the  body  solid  or  liquid  ?  When  gaseous  ?  What  was 
Davy's  view  ?  How  did  Clausius  differ?  Explain,  according  to  this  view,  tlie  solid,  liquid, 
and  gaseous  states.  In  what  respects  does  experience  confirm  it  ?  How  is  combustion 
explained?  When  is  a  body  heated  by  conduction  ?  What  is  the  theoretical  iirocess  ?  When 
is  a  body  heated  by  radiation  ?  Explain  the  process.  What  are  heat  rays  ?  What  facts  prove 
that  heat  and  light  rays  are  identical  ?  What  is  the  solar  specti-um  ?  What  different  rays  do 
we  distinguish  in  it?  What  physical  difference  is  there  between  the  heat,  light,  and  chemical 
rays  ?  What  physical  similarity  ?  What  facts  go  to  prove  that  it  is  the  motion  of  the  molecules 
of  a  body  ratlier  than  its  contained  ether  atoms  wbicli  constitute  its  heat?  When  is  a  body 
said  to  be  transparent?  Opaque?  Athermanous?  Diathermanous  ?  Are  the  heat  rays  emitted 
from  a  body  always  the  same  in  kind  ?  What  does  this  indicate  ?  What,  then,  seems  most 
probably  to  constitute  the  heat  of  a  body  ?  Are  good  radiators  of  heat  good  absorbers  ?  Why  ? 
Explain  now,  in  detail,  your  ideas  of  the  heating  of  a  body  by  a  distant  source,  and  the  processes 
of  radiation  and  conduction.  Do  these  or  any  other  theoretical  views  ailect  in  any  degree  the 
validity  of  our  general  principle  of  the  equivalence  of  heat  and  work  ?  Does  our  principle 
depend  upon  any  hypothesis  as  to  the  nature  of  heat  ?  Upon  what  does  it  depend  ?  Have 
these  views  any  value  ? 


CHAPTEE  III 

INNER  AND   OUTER  WORK. — LATENT  AND   SPECIFIC   HEAT. 

Whatever  may  be  tlie  motion  in  tlie  interior  of  a  body, 
whether  a  motion  of  ether  envelopes  according  to  the  views  of 
Eedtenbacher,  or  a  motion  of  the  molecules  as  assumed  by 
Clausius  and  most  other  physicists ;  whatever  may  be  the  char- 
acter of  the  motion,  this  much  at  least  is  established — that 
heat  is  some  sort  of  motion  of  the  particles,  atoms,  or  mole- 
cules. If  the  mass  of  a  particle  is  m,  and  its  velocity  v,  then 
^mv"  is  its,  living  force.  If  we  assume  the  entire  weight  of  the 
body  to  be  G,  then 

is  the  entire  inner  living  force  of  the  body,  or  its  entire  ''inher- 
ent energy,"  if  v  is  the  mean  velocity  of  an  atom. 

The  imparting  of  heat  to  the  body  has,  in  general,  three 
effects. 

l6'^.  TJie  temjjerature  of  the  body  rises — its  "sensible  heat"  is  in- 
creased. 

2id.   The  body  expands — its  volume  is  increased. 

3cZ.  In  this  expansion  the  exterior  pressure — generally  that  of  the 
atmosphere — is  overcome. 

Different  Worlds  performed  by  the  Heat — We  see,  then,  that  the 
heat  performs  a  threefold  work. 

1st.  Since  the  rise  of  the  temperature  consists  in  an  increase 
of  the  living  force  of  the  particles,  it  must  perform  a  work 
equivalent  to  this  increase. 

2d.  Since  the  molecules  mutually  attract  each  other,  a  cer- 
tain work  is  necessary  to  alter  their  mutual  positions  ;  or,  what 
is  the  same  thing,  to  move  their  common  center  of  gravity. 

3d  Since  the  body  is  pressed  upon  on  all  sides  by  the  air,  a 

121 


122  THERMODYNAMICS. 

certain  work  is  required  to  overcome  this  pressure  through  a 
certain  distance. 

That  work  which  goes  to  increase  the  velocity  of  the  mole- 
cules, and  therefore  to  increase  the  sensible  heat  of  the  body, 
we  call  the  "vibration  loorli,"  while  that  work  which  is  neces- 
sary to  displace  the  particles  we  call  the  "  disgregation  work." 

Outer  and  Inner  Work. — Since,  in  the  practical  applications  of 
the  mechanical  theory  of  heat,  that  work  necessary  to  overcome 
the  outer  pressure  is  of  especial  importance,  we  shall  follow 
Zeuner,  and  call  the  work  which  increases  the  velocity  of  vibra- 
tion, together  with  that  which  changes  the  aggregation  of  the 
particles,  that  is,  the  "vibration  work"  and  the  " disgregation 
work,"  the  "  i7in£r  tvork,"  while  we  shall  call  the  other,  which 
overcomes  the  outer  pressure,  the  "outer  ivork." 

Starting,  then,  from  the  experimentally  proved  law  that 
"  Heat  and  mechanical  work  are  equivalent,"  we  can  lay  down 
the  following  most  important  fundamental  principle  of  the  me- 
chanical theory  of  heat : 

The  amount  of  heat  [expressed  in  heat  units)  imparted  to  a  body 
is  directly  proportional  to  the  simultaneously  produced  inner  and 
outer  ivork. 

Specific  Volume — Specific  Pressure. — In  our  discussions  we 
shall,  in  general,  take  the  kilogram  (about  2.2  pounds)  as  the 
unit  of  weight.  The  volume  of  any  body  of  this  weight  we 
call  the  specific  volume,  and  shall  denote  it  by  v.  V  indicates 
the  volume  of  a  body  which  weighs  more  or  less  than  just  one 
kilogram.  The  pressure  upon  each  square  meter  of  surface 
of  any  body  we  call  the  specific  pressure,  and  shall  denote  it 
by  p.  We  therefore  assume  that  the  pressure  upon  each  unit 
of  surface  is  the  same.  We  also  assume  in  what  follows  that 
the  pressiire  upon  the  interior  of  the  surface  of  a  body,  at  any  in- 
stant, is  just  equal  to  the  outer  pressure,  or  varies  from,  it  by  an 
infinitely  small  amount.  That  is,  the  "  body  tension  "  is  at  any 
moment  just  equal  to  the  outer  pressure.  Whenever  we  de- 
viate from  this  assumption  we  shall  specially  indicate  it. 

[We  shall  do  well  to  distinguish  between  the  "  body  tension  "  and  the 
outer  pressure.  If  we  conceive  the  body  inclosed  by  a  tig-ht-fitting  envelope, 
or  skin,  the  tension  of  this  skin  is  the  body  tension.     This  may  or  may  not  be 


LATENT  AND  SPECIFIC  HEAT  123 

equal  to  the  external  pressure.     Unless  distinctly  stated  to  the  contrary,  we 

suppose  that  the  body  tension  at  any  moment  differs  from  the  outer  pressure 

only  by  an  indefinitely  small  amount,  and  hence  that  any  change  from  one 

state  to  another  is  continuous,  and  therefore  reversible.     The  outer  pressure  is 

thus  a  property  of  the  body. 

The  case  is  quite  dift'eient  when  we  assume  that  equilibrium  between  the 

outer  pressure  and  body  tension  does  not  exist  during  the  changes  of  condi- 

iion,  and  therefore  that  equilibrium  is  only  attained  at  the  beginning  and  end 

of  the  change,  when  the  body  has  passed  from  cue  condition  of  equilibrium  to 

another.     The  deportment  of  a  body  during  such  a  transition  is  evidently  of  a 

very  different  character,  and  we  are  able  to  follow  the  transition  only  in  a  few 

special  cases,  in  which,  by  certain  assumptions  upon  the  law  of  change  of  the 

outer  pressure  (which  is  considered  quite  independent  of  the  body  tension),  we 

are  able  to  determine  the  final  condition  of  the  body  after  the  occurrence  of  a 

new  state  of  equilibrium.     The  two  cases  may  be  represented  by  a  rod,  which 

in  the  first  case  is  stretched  by  a  force  at  any  instant  greater  only  by  an 

infinitely  small   amount  than  the  force  with  which  at  that  instant   the  rod 

resists  extension,  and  in  the  second  case,  acted  upon  suddenly  by  a  constant 

force.     In  the  first  case  the  rod  is  gradually  extended,  and  the  work  of  ex- 

Pl 
tension  can  be  easily  shown  to  be  -r-  ,  where  Q  is  the  final  force  and  I  is  the 

a 

extension.  In  the  second  case  it  can  be  shown  that  the  rod  is  elongated  twice 
as  far,  and  the  work  up  to  the  moment  of  greatest  elongation  is  2GI.  Further, 
the  end  of  the  rod  will  vibrate  up  and  down  like  a  pendulum  vdiose  length  is 
I,  and  finally  come  to  the  same  state  of  equilibrium  as  the  first. 

In  every  case,  therefore,  unless  otherwise  stated,  we  assume  that  during 
change  of  state  by  accession  or  withdrawal  of  heat  the  outer  pressure  is  equal 
at  any  instant  to  the  body  tension.  Thus  in  the  case  of  a  gas  expanding,  it 
overcomes  a  pressure  just  equal  to  its  own  tension  or  pressure  upon  the  piston 
at  any  instant.  When  this  is  the  case  the  change  of  state  is  always  reversible, 
i.  e.,  by  compression  the  gas  passes  through  all  intermediate  states  in  reverse 
order  back  to  original  state.] 

Fundamental  Equations  of  the  Meclianical  Theory  of  Heat. — If, 
now,  we  impart  to  tlie  unit  of  weiglit  of  any  body  one  lieat  unit 
(that  is,  so  mucli  lieat  as  will  raise  tlie  temperature  of  one 
kilogram  of  2vater  from  0°  to  1°  C),  we  increase  its  total  energy 
by  424  meter-kilograms ;  that  is,  tlie  increase  of  its  inner  and 
outer  work  is  equivalent  to  a  work  of  424  meter-kilograms. 
If,  however,  we  impart  Q  heat  units,  then  the  work  performed 
upon  the  body,  or  received  by  it,  is  424  x  Q.  If  we  denote 
this  work  by  E,  we  have  for  the  energy  imparted  to  each  unit 
in  weight  by  the  reception  of  Q  units  of  heat, 

^  =  424  X  Q, 
and  inversely, 


124  THERMODYNAMICS. 

Tlie  fraction  4I4,  or  that  portion  of  one  lieat  unit,   or  tliat 
amount  of  lieat  wMch  is  equivalent  to  one  unit  of  work,  we 
designate  in  general  by  ^4. 
We  thus  have 

Q^  A->i  E. 

If,  now,  W  is  that  portion  of  tlie  whole  energy  which  goes  to 
the  increase  of  the  vibration  work,  J  that  portion  which  goes 
to  disgregation  work,  and  L  that  portion  which  corresponds  to 
the  outer  work,  we  have  E—W+  J  +  L,  and  hence 

Q=A{W+J+L) (I.) 

Since  we  call  both  W  and  J  together  the  inner  work,  let  us 
represent  it  by  U,"  or  Z7 --  W+  J,  and  we  then  have 

Q=A{U+L) (II.) 

Change  of  Sign  of  the  Terms  in  Equation  I. — Now  in  Equation  I., 
any  one  or  more  of  the  terms  in  the  parenthesis  may  be  zero, 
or  may  be  negative.  In  such  case,  for  the  same  amount  of 
heat  imparted,  Q,  the  remaining  terms  must  be  greater.  We 
may  illustrate  by  a  few  examples. 

It  has  been  already  remarked,  that,  in  order  to  convert  ice  at 
0^  into  water  at  0^,  not  less  than  about  80  heat  units  are  neces- 
sary. When,  therefore,  we  add  heat  to  the  ice,  we  do  not 
increase  its  vibration  work,  i.  e.,  its  sensible  temperature  at  all, 
but  the  heat  imparted  performs  disgregation  work  and  outer 
work.  In  Equation  I.,  therefore,  W  is  zero.  This  is  the  case 
with  almost  all  other  bodies.  The  heat  which  thus  disappears, 
apparently,  since  it  is  not  sensible  to  the  thermometer  or  to 
our  nerves  while  a  body  is  melted,  we  call  the  "latent  heat." 

The  experiments  of  Eegnault  and  Person  give  the  latent  heat 
of  several  bodies  as  follows  : 

Water 79.25         Tin 14252        Zinc 28.13 

Phosphorus. 5.034        Lead 5.369        Mercury   .2.83 

Sulphur 9.368         Bismuth..  12. 640 

In  passing  from  the  liquid  to  the  gaseous  state  heat  also 
disappears,  and  the  amount  thus  disappearing  is  in  general 
greater  than  in  liquefaction. 

*  [Kirohoff  calls  the  quantity  ZJthe  "  working  function  ;"  Thomson,  "  the  mechanical  energy 
of  a  body  in  a  given  state  ;"  Clausius  understands  by  "  inner  work"  only  tliat  portion  denoted  by 
J,  which  we  call  "  disgregation  work."] 


LATENT  AND  SPECIFIC  HEAT.  125 

Thus,  for  example,  according  to  Brix,  the  latent  heat  of  steam 
is  540 ;  that  is,  to  convert  one  kilogram  of  water  at  100^  into 
steam  at  100,  540  heat  units  are  necessary;  that  is.  Just  as 
much  heat  as  is  required  in  order  to  heat  540  kilograms  of 
water  from  0'  to  1'.  Thus,  referring  to  theory,  a  greater  ex- 
penditure of  work  is  required  to  give  the  particles  a  rectilinear 
motion,  such  as  we  assume  for  gas,  than  to  impart  those  rotary 
motions  which  are  supposed  to  constitute  a  liquid. 

For  alcohol,  sulphuric  ether,  and  oil  of  turpentine,  Brix  found 
the  latent  heat  of  evaporation,  210,  89.96,  and  74.04,  respec- 
tively. 

If  we  heat  water  from  0"  to  4^  it  does  not  expand,  as  is  gen- 
erally the  case  when  bodies  are  heated,  but  contracts,  and  thus 
the  molecules  approach  each  other.  The  outer  work  is  thus 
negative.  In  fact  the  outer  pressure  here  assists  the  heat,  so 
to  speak.  Without  this  pressure  we  would  have  to  impart 
more  heat  in  order  to  raise  the  same  water  up  to  4".  If,  there- 
fore, we  increase  the  pressure,  the  heat  necessary  to  be  im- 
parted is  less. 

We  have  still  greater  contraction  when  we  convert  ice  at  O'' 
into  water  at  0  \  Ermann  observed  that  the  volume  of  water 
at  0'  is  only  fVths  of  that  of  the  ice  at  the  same  temperature. 
Here,  then,  the  outer  work  has  a  still  greater  negative  value, 
and  therefore,  under  increased  pressure,  the  latent  heat  of  the 
ice  ought  to  be  diminished,  i.  e.,  its  melting  point  lowered. 
Mayer  predicted  this  in  his  contributions  to  the  mechanical 
theory  of  heat,  and  the  experiments  of  Mousson  have  com- 
pletely confirmed  it.  This  physicist  showed  by  a  very  ingen- 
ious experiment  that  under  a  pressure  of  about  13,000  atmos- 
pheres the  melting  point  of  ice  was  lowered  about  18°,  that 
therefore,  under  this  pressure,  ice  became  liquid  at  —  18°, 
instead  of  0°. 

Cast  iron  and  bismuth  also  contract  when  they  pass  from  the 
solid  to  the  liquid  condition.  The  most  striking  example  of 
contraction  when  heated  is,  according  to  Sir  William  Thom- 
son, shown  by  vulcanized  rubber.  When  a  tube  of  this  material 
is  fastened  at  one  end,  and  a  weight  of  ten  or  more  pounds  is 
hung  from  the  other,  this  weight  is  raised  by  heating  the  tube. 
We  see  here,  therefore,  very  plainly,  that  the  outer  work  per- 
formed is  negative. 


MELTING  POINT 

Spermaceti. 

Wax. 

Sulphur. 

Stearine. 

51°  C. 

645°  C. 

107°  C. 

72.5°  C. 

60^ 

74.5° 

135.2° 

73.6° 

80.2° 

80.2° 

140.5° 

79.2° 

126  THERMODYNAMICS. 

Increased  Pressure  can  also  Raise  the  Melting  Point. — If  a  body 
expands  under  the  action  of  lieat,  this  expansion  will  be  less 
wlien  tlie  pressure  is  increased,  if  the  heat  imparted  is  the 
same  in  amount.  And  inversely,  if  the  expansion  is  the  same, 
more  heat  must  be  imparted.  If  a  body  in  melting,  then,  fol- 
lows the  general  rule  and  expands,  its  melting  point  must  rise 
when  it  is  subjected  to  a  greater  pressure.  In  this  case  the 
outer  work,  L  performed  during  expansion,  is  greater,  and 
accordingly  the  heat  imparted,  Q,  must  be  also  greater.  These 
theoretical  conclusions  are  also  confirmed  by  numerous  experi- 
ments. Thus  Hopkins  has  found  for  the  melting  point  of  vari- 
ous substances,  under  varying  pressures,  the  following  results  : 

Pressure 

in  At- 
mospheres. 

1 

519 

792 

Heat  luMcJi  must  he  Imjoarted  to  Gases  under  different  Conditions 
for  equal  Rise  of  Temperature. — In  the  case  of  gases,  whose  mole- 
cules are  not  bound  together  by  mutual  attractions,  no  work 
is  expended  in  changing  the  state  of  aggregation.  When,  there- 
fore, we  impart  heat  to  a  gas,  the  vibration  work  is  increased, 
and,  under  certain  circumstances,  outer  work  is  performed.  It 
is  not  difficult  here  to  do  away  with  the  outer 
-Q  work  also,  and  then  a  less  quantity  of  heat  is 
necessary  to  heat  the  gas  a  certain  number  of 
degrees  than  when  outer  work  is  performed  dur- 
ing the  expansion.  This  may  be  illustrated  by 
the  following  experiment : 

Suppose   that  air  is  inclosed    in  the    cylinder 
^     ABCD,  Fig.  9,  below  the   air-tight   piston  EF. 
When  the  air  is  heated  it  expands  and  raises  the 
piston.     The  heat  imparted  then  goes  to  increase 
^      the  vibration  work  and  to  perform  outer  work. 
Let  us  now  make  the  piston  fast,  so  that  expan- 
sion is  impossible ;  then  the  outer  work  is  zero, 
and  less  heat  is  necessary  in  order  to  raise  the  air  the  same 
number  of  degrees  as  before. 


D 


LATENT  AND  SPECIFIC  HEAT.  127 

We  can  easily  make  here  tlie  outer  work  negative.  "We 
have  only  to  suppose  that,  while  the  air  is  being  heated, 
weights  are  gradually  applied  to  EF,  so  that,  in  spite  of  the 
rise  of  temperature,  the  piston  sinks,  and  the  air  is  com- 
pressed. The  work  performed  thus  by  the  sinking  piston  cor- 
responds to  a  certain  amount  of  heat,  and  the  whole  amount 
imparted,  in  order  to  raise  the  air  to  the  same  temperature, 
is  less  than  before  by  just  this  amount. 

Change  of  the  Fundamental  Equations  ivhen  Heat  is  Abstracted. — 
If  we  take  heat  from  a  body,  inverse  phenomena  occur ;  the 
vibration  work  is  in  general  diminished,  and  there  is  there- 
fore a  decrease  of  temperature  ;  also  the  molecules  approach 
nearer,  and  the  outer  work  is  negative.  Equation  I.  accord- 
ingly takes  the  form 

-  Q^A{-  W-J-L). 

Just  as  in  Equation  I.  different  terms  in  the  parenthesis  can  be 
zero  or  negative,  so  here  different  terms  can  be  zero  or  pos- 
itive. 

"We  have  an  example  in  the  case  of  Avater.  When  this  is 
cooled  from  4°  to  0"  it  does  not  contract,  but  expands,  the  outer 
work,  L,  is  therefore  positive.  This  positive  outer  work  is  still 
greater  when  we  abstract  from  the  water  at  0°  its  latent  heat  of 
liquefaction,  and  thus  convert  it  into  ice  at  0°.  In  this  case 
the  vibration  work,  or  sensible  heat,  remains  the  same,  and 
hence  W  =  0.  Since,  now,  W  =  0,  and  L  is  positive,  J  must 
have  a  so  much  greater  negative  value,  that  is,  the  molecules 
are  the  more  strongly  attracted.  We  can  explain  this  only  by 
assuming  that  the  molecules  are  arranged  in  a  definite  and 
regular  manner,  that  is,  that  the  ice  has  a  crystalline  consti- 
tution. 

As  in  the  freezing  of  most  other  bodies  there  is  a  contraction, 
the  outer  work  is  negative  when  the  latent  heat  of  liquefaction 
is  abstracted,  and  hence  the  decrease  of  the  vibration  work  is 
zero.  For  these  bodies,  for  the  same  withdrawal  of  heat,  the 
attraction  of  the  molecules  increases  relatively  less  rapidly  than 
for  water. 

Also,  when  steam  loses  its  latent  heat  of  vaporization  and 
becomes  liquid,  the  decrease  of  its  vibration  work,  or  its  sen- 


128 


TEEBMOD  YNAMIC8. 


II 


D 


B' ..  !il!iaiMmi 


II 


sible  heat  is  zero.  As  we  say,  tlie  steam  is  condensed,  since 
steam  occupies  a  very  miicli  greater  space  tlian  the  liquid  from 
which  it  is  generated,  the  outer  work,  L,  must  have  a  very 
great  negative  value. 

This  work  Papin  sought  to  utilize.  The  cylinder  ABCD, 
Fig.  10,  contained  water,  which  was  heated  until  all  the  air  was 
driven  out.  Then  the  air-tight  piston 
was  inserted,  and  cold  water  a23plied 
to  the  cylinder,  thus  condensing  the 
steam.  The  atmospheric  pressure 
then  forced  the  piston  HH  down, 
and  thus  raised  the  rod  /  by  means  of 
the  lever  FG.  "When  the  piston  ar- 
rived at  the  bottom  the  water  w^as 
again  heated,  the  expansive  force  of 
the  steam  balanced  the  air  pressure, 
and  the  weight  of  I,  which  was  heavier 
than  the  piston,  then  raised  the  lat- 
ter to  the  top  of  the  cylinder.  The  steam  was  then  again  con- 
densed, and  so  on. 

As  is  well  known,  Newcomen  first  practically  utilized  this 
idea  in  England,  in  his  atmospheric  engine,  w^hich  Watt  later 
converted  into  the  steam  engine. 

This  process,  by  which  a  body,  as  the  water  in  this  case,  is 
changed  from  one  condition  to  another,  and  from  this  back 
again  to  the  first,  is  called  a  "  cycle  process^ 

We  have,  also,  in  hot-air  engines  a  similar  cycle  process,  but 
instead  of  water  it  is  air  which  is  made  to  change  its  condition. 
First,  a  certain  volume  of  air  is  compressed,  then  heated,  and 
thus  outer  work  is  performed,  then  cooled  back  to  its  original 
condition,  and  so  on. 


Specific  Heat — If  we  impart  equal  amounts  of  heat  to  differ- 
ent bodies,  the  increase  of  vibration  work,  that  is,  of  their  sen- 
sible heat,  is  very  different. 

We  may  illustrate  this  by  an  experiment.  If  we  mix  one 
kilogram  of  water  at  10°  with  one  at  30",  the  temperature  of  the 

.     1  X  10  +  1  X  30      „ .  o      T. 

mixture  is  ^r =  20  .     For,  since  we  require,  m 

the  one  case,  to  raise  one  kilogram  of  water  to  10°,  1  x  10  =  10 


LATENT  AND  SPECIFIC  HEAT.  129 

heat  units,  and,  in  the  other,  1  x  30  =  30  heat  units,  there  are 
required  for  both  10  +  30  =  40  heat  units,  or  V  =  20°  for  each 
kilogram. 

If,  now,  we  mix  one  kilogram  of  water  at  10°  with  one  kilo- 
gram of  iron  at  30°,  we  shall  find  that  the  temperature  of  the 
mixture  is  12°.  Thus  the  iron  has  lost  30  —  12  =^  18°,  and  this 
heat  has  been  imparted  to  the  water.  But  this  heat  does  not 
raise  the  kilogram  of  water  18°,  but  only  2°.  What  the  iron 
has  lost  is,  apparently,  not  gained  by  the  water. 

If,  on  the  other  hand,  we  mix  one  kilogram  of  water  at  30° 
with  one  kilogram  of  iron  at  10°,  we  shall  find  the  temperature 
of  the  mixture  to  be  28°. 

The  2°,  or  the  2  heat  units  lost  by  the  water,  thus  raise  the 
temperature  of  the  iron  18°.  One  heat  unit,  therefore,  will 
raise  one  kilogram  of  iron  9°. 

We  see,  therefore,  that  the  same  amount  of  heat  which 
causes  in  water  a  certain  rise  of  temperature,  has  an  effect  nine 
times  as  great  for  iron. 

If,  again,  we  have  one  kilogram  of  water  at  about  80°  (more 
exactly  79.25°),  it  will,  as  we  know,  render  completely  liquid 
one  kilogram  of  ice  at  0° ;  and,  on  the  other  hand,  with  one 
kilogram  of  iron  at  80°  we  can  melt  only  ^th  of  a  kilogram  of  ice 
atO°. 

Also,  if  we  allow  equal  quantities  of  water  and  iron,  at  the 
same  temperature,  to  cool  in  the  air,  we  shall  find  that  the 
temperature  of  the  iron  sinks  nine  times  as  fast  as  that  of 
the  water.  But  since  the  air  abstracts  from  both  bodies,  in 
the  same  time,  equal  quantities  of  heat,  the  water  must  pos- 
sess nine  times  as  much  heat,  at  the  same  temperature,  as  the 
iron. 

Experiments  have  shown  that  two  bodies  seldom  occur  for 
which  equal  weights  are  raised  by  equal  amounts  of  heat  the 
same  number  of  degrees. 

We  call  that  amount  of  heat,  expressed  in  heat  units,  which 
is  necessary  to  raise  one  kilogram  of  a  body  one  degree,  the 
"specific  heat^'  of  the  body. 

In  the  following  table  we  have  the  specific  heat  of  various 
bodies,  as  determined  under  constant  pressure,  by  the  exact  and 
careful  experiments  of  Eegnault : 
9 


130  THERMODYNAMICS. 


Antimony 0.0508 

Bismuth 0.0308 

Carbon 0.2414 

Cobalt 0.1070 

Copper 0.0952 

Gold 0.0334 

Iron 0.1138 

Lead 0.0314 

Manganese 0.1217 


Mercury 0.0333 

Nickel 0.1110 

Phosphorus 0.1887 

Platinum 0.0324 

Silver 0.0570 

Sulphur 0.2026 

Tin 0.0562 

Zinc 0.0956 


Volume  Capacity.— The  specific  heat  defined  above  is  the 
quantity  of  heat  expressed  in  heat  units  necessary  to  raise  one 
kilogram,  or  one  icnit  of  iveiglit  of  a  body,  one  degree.  "We  may 
therefore  call  it  the  heat  capacity  of  one  unit  of  weight,  or  gen- 
erally, the  "  iveiglit  capacity.'''' 

But  we  may  also  easily  determine  the  quantity  of  heat  re- 
quired to  raise  equal  volumes  of  different  bodies  one  degree. 

One  kilogram  of  water  occupies  a  space  of  one  cubic  deci- 
meter, since  one  cubic  meter  of  water  weighs  1,000  kilograms. 
Now,  the  density  of  chemically  pure  iron  is  7.8439  times  as 
great  as  that  of  water,  hence  one  cubic  decimeter  of  iron  weighs 
7.8439  kilograms. 

Since,  now,  one  kilogram  of  iron  requires  0.1138  heat  units 
to  raise  its  temperature  one  degree,  7.8439  kilograms,  or  one 
cubic  decimeter,  requires  7.8439  x  0.1138  =  0.8926  heat  units. 
This  quantity  of  heat  we  may  call  the  "  volume  capacity  "  of  the 
iron.  In  general,  we  understand  by  volume  capacity  of  a  body 
that  amount  of  heat  necessary  to  raise  equal  volumes  one  degree  in 
•  temperature. 

We  may  obtain  it,  as  shown  by  the  example  above,  hy  multi- 
plying tJie  specific  gravity  of  the  body  hy  its  weight  capacity,  or  its 
"specific  heat." 

[We  owe  the  latest  researches  upon  the  specific  heat  of  gases  to  Regnault ;' 
but  these  researches  give  only  the  specific  heat  for  constant  pressure,  that  for 
constant  volume  has  not  been  as  yet  directly  determined.  Let  Cp  be  the  weight 
capacity  for  constant  pressure,  and  w,,  the  volume  capacity  for  constant  press- 
ure, which,  as  we  have  seen  above,  can  be  found  from  the  weight  capacity  by 
multiplying  by  the  specific  weight.     Then  we  have 

C;,.  Wp  . 

Air 0.23751  0.00030714 

Nitrogen 0.24380  0.00030625 

Oxygen 0.31751  0.00031099 

Hydrogen 8.40900  0.00030533 


LATENT  AND   SPECIFIC  HEAT.  131 

The  values  of  tlie  weight  capacity  for  constant  pressure  are,  as  we  see,  dif- 
ferent for  different  gases.  Hydrogen  is  most  noticeable.  Its  specific  heat  is 
inded  greater  than  for  any  other  body,  solid  or  liquid.  After  hydrogen  comes 
water,  whose  specific  heat  {c,)  is  1.  For  others  the  specific  heat  for  constant 
volume  is  less,  and  indeed  for  most  much  less  than  1. 

Regnault  also  found  the  specific  heat  of  gases,  especially  of  air,  constant  for 
different  pressures  and  temperatures,  a  beautiful  confirmation  of  a  prediction 
made  by  Clausius,  as  early  as  1850,  upon  theoretical  grounds  alone. 

As  regards  the  volume  capacities  for  constant  pressure,  we  see  that  they 
differ  but  little,  so  little  that  one  is  inclined  to  assert  that  the  deviations  are 
due  merely  to  errors  of  observation.  If  we  calculate,  however,  the  volume 
capacity  for  the  other  gases  given  by  Regnault,  we  find  that  approximate  equal- 
ity exists  only  for  those  gases  which  are  furthest  from  their  point  of  liquefac- 
tion, and  in  which,  therefore,  the  disgregation  work  is  of  little  or  no  account. 
We  can,  therefore,  conclude  that  equality  exists  only  for  "  perfect"  gases,  and 
this  conclusion  is  supported  by  theory. 

As  to  the  specific  heat  of  gases  for  constant  volume,  as  already  remarked,  a 
direct  determination  is  not  yet  attained.  We  can,  however,  determine  this 
value  indirectly,  at  least  for  air.  If  c,,  is  the  weight  capacity  of  air  for  con- 
stant pressure,  and  c„  for  constant  volume,  we  can  in  various  ways  determine 

the  ratio  Ic  =  —.    Such  a  determination  we  shall  hereafter  give  when  we  come  to 

apply  the  principles  of  the  mechanical  theory  of  heat  to  the  solution  of  different 
problems.  One  of  these  methods,  used  by  Gay-Lussac,  Clement,  and  Desormes, 
and  later  by  Masson  {Wullner,  Experimental  Physik,  Bd.  2,  p.  279),  gave  re- 
spectively k  =  1.372,  1.357,  and  1.419.  Another  method,  by  Hirn  {Theorie  me- 
canique  de  la  cJialeur,  p.  69),  atid  Weisbach  {CivUingenieur,  Bd.  5,  p.  46),  gave 
1.3845  and  1.4025.  Further,  a  comparison  of  the  results  of  the  formuljs  for  the 
velocity  of  sound,  by  Dulong,  with  observations  upon  the  progression  of  sound 
in  air,  gave  k  — 1.421,  and  Dulong  found  similar  values  also  for  nitrogen,  oxy- 
gen, and  hydrogen.  A  similar  comparison,  with  the  results  of  observation  by 
Moll  and  Van  Bech  upon  the  velocity  of  sound,  gave  for  air  k  =  1.410. 

This  last  value  is  regarded  at  present  as  the  most  reliable,  especially  as  it  is 
justified  by  the  results  of  other  researches. 

Taking  for  air,  then,  k  =  -  -=  1.41,  and  c^,  according  to  Regnault  0.28751, 

we  have  for  the  specific  heat  of  air  for  constant  volume  c,  =  0.16844. 

Since  the  outer  pressure  may  be  supposed  indefinitely  varied,  there  may  be, 
strictly  speaking,  an  indefinite  number  of  values  for  the  specific  heat  of  a  body, 
one  for  each  different  law  of  variation  of  pressure.  We  have  considered  above 
only  two  special  cases,  viz.,  for  constant  pressure  and  for  constant  volume. 
We  shall  see  later  how  to  find  the  specific  heat  for  any  given  law  of  variation 
of  pressure  with  volume.  In  Regnault's  experiments  the  bodies  were  sub- 
jected to  the  constant  pressure  of  the  atmosphere,  and  in  accepting  his  results 
we  must  not  therefore  neglect  the  fact  that  they  strictly  hold  good  only  for 
constant  outer  pressure. 

For  solid  and  most  liquid  bodies  the  expansion  is  very  slight,  and  therefore 
the  heat  converted  into  work  insignificant  in  such  case.  For  gases,  however, 
as  we  have  seen,  the  difference  is  great. 


132  THERMODYNAMICS. 

It  is  evident  that  the  specific  heat  for  constant  pressure  Cp  must  alwiiys  be 
greater  than  that  for  constant  volume  Cj,  because  in  the  first  case  heat  is  ab- 
sorbed by  the  outer  work,  and  for  the  same  rise  of  temperature  more  heat  must 
be  imparted. 

In  general,  vpe  have  for  the  heat  imparted  for  a  rise  of  temperature  at, 
for  the  unit  of  volume  heated  under  constant  volume,  in  vphich  case  dL  the 
outer  work  is  zero, 

dQ„  =  codt  +  AdJ. 

For  a  perfect  gas  dJ,  or  the  increase  of  disgregation  work,  is  zero,  and  then 
oo  is  the  specific  heat  of  the  unit  of  volume  for  constant  volume  (volume  capac- 
ity). If,  however,  J  is  not  zero,  then  the  value  which  we  have  until  now  called 
the  volume  capacity  has  a  complex  significance.  Without  doubt  dJ  is,  under 
the  above  conditions,  positive,  since  we  must  admit  that  the  smallest  particles 
act  attractively  upon  each  other,  and  dJ  represents  the  work  which  even  by 
constant  volume  is  applied  to  disgregation  of  the  molecular  groups.  Rankine 
calls  the  value  of  oo  in  the  above  equation  the  "  real  specific  heat,"  and  hence 
we  should  conclude  that  the  "  apparent  specific  heat "  deviates  the  more  from 
the  real,  and  is  so  much  greater  than  it,  the  more  the  gas  departs  from  the 
perfect  condition. 

The  total  heat,  then,  imparted  in  order  to  raise  a  body  one  degree  in  tem- 
perature is  the  apparent  specific  heat ;  if  from  this  be  subtracted  all  the  heat 
expended  in  performing  interior  and  exterior  work,  the  remainder  is  the  real 
specific  heat,  because  it  alone  measures  the  actual  heat  of  the  body.] 

The  Disgregation  Work,  in  Solid  and  Liquid  Bodies,  is  very 
smalt  in  comparison  with  the  Vibration  Work. — We  liave  already 
repeatedly  defined  the  "  unit  of  lieat  "  as  that  quantity  of  heat 
which  must  be  imparted  to  one  kilogram  of  water  in  order  to 
raise  its  temperature  one  degree.  But  now,  if  we  impart  to,  say 
one  kilogram  of  water,  so  much  heat  that  its  temperature  is 
raised  one  degree,  we  have  not  only  increased  the  vibration 
work  or  sensible  heat,  but  also  the  disgregation  work  as  well. 
The  heat  imparted  must  therefore  be  greater  than  it  would 
have  been  had  we  only  increased  the  sensible  heat  of  the  water. 
Of  this  fact  no  account  was  taken  when  the  idea  of  the  heat 
unit  was  first  formed  and  the  specific  heats  of  bodies  deter- 
mined, because  at  that  time  the  principles  of  the  mechanical 
theory  of  heat  were  unknown.  If,  then,  the  disgregation  work 
were  considerable  and  varied  much  for  different  bodies,  these 
determinations  of  specific  heats  would  have  little  or  no  value. 
Fortunately  this  is  not  the  case.  Thus  Eegnault,  whose  ex- 
periments are  the  most  reliable,  made  his  observations  upon 
bodies  subjected  only  to  the  pressure  of  the  atmosphere.     The 


LATENT  AND  SPECIFIC  HEAT.  133 

outer  work  and  disgregation  work  tlius  performed  were  but 
sliglit  in  comparison  with  tlie  vibration  work. 

For  example,  let  us  suppose  one  kilogram  of  water  at  4°,  that 
is  at  its  greatest  density,  contained  in  a  cylindrical  vessel  of  one 
square  decimeter  in  cross-section.  The  depth  of  the  Avater  is 
then  exactly  one  decimeter.  If  this  water  is  raised  one  degree 
in  temperature  it  expands,  according  to  the  elaborate  experi- 
ments of  Kopp,  about  0.000006  of  its  volume  at  4°.  The  water 
thus  rises  in  the  vessel  about  0.000006  of  a  decimeter,  and 
through  this  distance  the  atmospheric  pressure  of  about  103.34 
kilograms  j)er  square  decimeter  is  overcome.  We  have  there- 
fore the  outer  work  equal  to  103.34  x  0.000006  =  0.00062004  deci- 
meter kilograms.  But  since  the  mechanical  equivalent  of  the 
heat  imparted  is  424  meter  kilograms,  the  outer  work  is,  as  we 
see,  very  small  in  comparison.  The  disgregation  work,  or  the 
work  expended  in  separating  the  particles,  is  also  very  small. 
Thus,  according  to  the  experiments  of  Grassi,  the  coefficient  of 
compression  of  water,  that  is,  the  amount  it  is  compressed  by 
an  increase  of  pressure  of  one  atmosphere,  is  only  0.00005  of 
its  volume  when  the  compression  takes  place  between  4°  and  5°. 
If,  then,  we  increase  the  pressure  upon  the  water  in  our  vessel 
by  one  atmosphere,  the  water  is  compressed  about  0.00005  of  a 
decimeter.  This  corresponds  to  a  work  therefore  of  0.00005 
X  103.34  =  0.005167  decimeter  kilograms.  We  see  thus  that 
the  disgregation  work  is  also  very  small.  We  may,  therefore, 
for  slight  rise  of  temperature,  disregard  the  entire  disgregation 
work  for  solid  and  liquid  bodies. 

Since  for  high  temperatures  the  coefficient  of  expansion  in- 
creases, that  is,  the  increase  of  volume  for  a  rise  of  1°  is  greater, 
the  disgregation  work  must  increase,  and,  for  the  same  amount 
of  heat  imparted,  the  vibration  work  or  sensible  heat  be  pro- 
portionally less.  If  the  increase  of  this  last  is  the  same  as 
before ;  that  is,  if  the  body  at  the  higher  temperature  is  raised 
also  1°,  then  the  total  heat  imparted  is  greater  than  before. 
We  see,  therefore,  the  reason  why  the  s'pecific  heat  increases  with 
the  temperature. 

It  is  also  a  fact  that  liquid  bodies  expand  more  for  the  same 
rise  of  temperature  than  solid.  It  is  therefore  very  probable 
that  the  same  body  has  a  greater  coefficient  of  expansion  when 
in  the  liquid  condition  than  when  solid ;  and  this  renders  it 


134  THERMODYNAMICS. 

also  probable  that  the  specific  heat  of  a  body  ivhen  liquid  is  greater 
than  when  solid.  For  many  bodies  this  lias  already  been  con- 
firmed by  careful  experiments. 

Specific  Heat  for  Constant  Volume  and  for  Constant  Pressure. — 
If  we  conceive  a  body  so  confined  that  it  cannot  expand  when 
it  is  heated,  then  all  the  heat  imparted  goes  to  increase  the 
vibration  work,  and  there  is  no  disgregation  work  or  outer 
work  at  all.  For  the  same  amount  of  imparted  heat,  therefore, 
the  sensible  heat  will  be  greater  than  when  the  body  is  free  to 
expand.  Inversely,  a  less  amount  of  heat  would  be  necessary 
to  cause  a  certain  rise  than  when  free  to  expand. 

The  amount  of  heat  measured  in  heat  units  which  must  be 
imparted  in  order  to  raise  one  kilogram  weight  of  any  body 
1°  in  temperature  when  expansion  cannot  take  place,  is  called 
the  "specific  heat  for  constant  volume.'" 

We  call,  on  the  other  hand,  that  amount  of  heat  which  is 
necessary  to  raise  the  same  unit  weight,  one  kilogram  of  any 
body,  1°  in  temperature,  when  the  body  is  allowed  to  expand 
under  the  constant  pressure  of  the  atmosphere,  the  "sjjecific 
heat  for  constant  pressure."  It  must  evidently  be  greater  than 
the  first,  because  heat  is  required  to  perform  the  disgregation 
work  and  outer  work  which  take  place  in  the  second  case. 

[Both  these  specific  heats  refer  to  the  unit  of  loeigM,  and  are  therefore 
"  weight  capacities,"  the  one  under  constant  volume  and  the  other  under  con- 
stant pressure.  We  might  also  have  two  "volume  capacities"  in  the  same 
circumstances.  No  use  is  made  of  such  quantities.  "  Specific  heat "  refers 
always  to  the  %init  of  imigTit,  and  indeed,  unless  distinctly  stated, -we  always 
understand  a  constant  volume  to  be  presumed.] 

[The  following  tables  give  tl)e  mean  specific  heats  for  constant  pressure  of 
the  substances  named,  according  to  Regnault.  These  specific  heats  are  average 
values,  taken  at  temperatures  which  usually  come  under  observation  in  tech- 
nical applications.  The  actual  specific  heats  of  all  substances,  in  the  solid  or 
liquid  states,  increase  slowly  as  the  body  expands  or  as  the  temperature  rises, 
and  when  great  accuracy  is  required  tables  of  specific  heats  must  be  consulted, 
which  will  give  these  quantities  with  greater  definiteness  at  special  tem- 
peratures. 


SOLIDS 

Antimony 0.0508 

Copper 0.0951 

Gold 0.0324 

Wrought  Iron 0.1138 


Steel  (soft) 0.1165 

Steel  Oiard) 0.1175 

Zinc 0.0956 

Brass 0.0939 


LATENT  AND   SPECIFIC  HEAT. 
SOLIDS —  Continued. 


135 


Glass 0.1937 

Cast  Iron 0.1298 

Lead 0.0314 

Platinum 0.0324 

Silver 0.0570 

Tin 0.0563 


Ice 0.5040 

Sulphur 0.2026 

Charcoal 0.2410 

Alumina 0.1970 

Phosphorus 0.1887 


LIQUIDS. 


Water 1.0000 

Lead  (melted) 0.0402 

Sulphur  "        0.2340 

Bismuth  "        0.0308 

Tin  "        0.0637 

Sulphuric  Acid 0.3350 


Mercury 0.0338 

Alcohol  (absolute) 0.7000 

Fusel  Oil 0.5640 

Benzine 0.4500 

Ether 0.5034 


Constant  Pressure.  Constant  Volume. 

Air 0.23751     0.16847 

Oxygen 0.21751     0.15507 

Hydrogen 3.40900 2.41226 

Nitrogen 0.24380     0.17273 


Superheated  Steam 0.4805 

Carbonic  Acid 0.217 

Olefiant  Gas 0.404 

Carbonic  Oxide 0.2479 

Ammonia 0. 508 

Ether 0.4797 

Alcohol 0.4534 

Acetic  Acid 0.4125 

Chloroform 0.1567 


0.346- 

0.1535 

0.173 

0.1758 

0.299 

0.3411 

0.3200 


QUESTIONS  FOE  EXAMINATION. 

Whatever  may  be  the  views  held  as  to  the  nature  of  hent,  what  have  we  thus  far  established  ? 
Is  this  firmly  established  ?  What  are  the  proofs  ?  What  do  j'ou  understand  by  "  living  force  ?  " 
What  by  "inherent  energy  ?  "  If  a  body  of  mass  m  has  a  velocity  ?',  what  work  can  it  perform 
in  coming  to  rest  while  overcoming  resistance  ?  Is  the  setting  of  a  body  in  motion  "  work  ?  " 
If  heat  is  a  motion  of  bodies,  is  it  not  then  identical  with  work  ?  If  it  is  not  such  motion, 
should  we  expect  to  find  any  equivalence  ?    Do  we  find  any  equivalence  ?    What  is  it  ? 

When  a  body  is  heated,  what  three  efEects  are  in  general  produced  ?  In  what  does  the  rise  of 
temperature  consist  ?  What  three  works  are  performed  when  a  body  is  heated  ?  What  do  you 
understand  by  vibration  work  ?  Disgregation  work  ?  Outer  work  ?  Inner  work  ?  How  do  we 
measure  heat  ?  What  is  a  "  heat  unit  ?  "  To  what  is  the  amount  of  heat  imparted  to  a  body 
proportional  ?  How  many  heat  units  are  equivalent  to  2120  meter-kilograms  ?  How  many  meter- 
kilograms  are  equivalent  to  12  heat  units  ?    What  does  A  denote  in  our  notation  ?    If  ^  is  work 


136  THERMODYNAMICS. 

expressed  in  meter-kilo?rauis,  and  Q  is  amount  of  heat  given  in  heat  units,  what  is  the  general 
equation  between  them  ?  What  is  the  general  relation  between  Q  and  the  vibration  work  (IF), 
disgregation  work  (.7),  and  outer  work  (L)  ?    What  does  C/" denote  in  our  notation  ? 

Define  "specific  volume  "—"  specific  pressure."  What  does  v  denote?  What  does  F  de- 
note ?  What  do  you  understand  by  "body  tension?"  What  is  the  relation  between  body 
tension  and  outer  pressure  which  is  assumed  unless  otherwise  stated  ? 

In  tlie  equation  Q=  A  {W-\-J  -\-  L),  what  do  the  letters  denote  ?  Give  a  familiar  example 
where  Wis  zero.  What  other  examples  can  you  think  of?  Define  "latent  heat?"  What  is 
the  latent  heat  of  steam  ?  Give  a  familiar  example  in  which  L  is  negative.  Can  increased  press- 
ure raise  the  melting  point?  Under  what  circumstances  is  this  true,  and  why  ?  Give  a  familiar 
example  where  J  is  zero. 

When  a  gas  is  heated,  what  works  are  performed  ?  When  a  gas  is  not  free  to  expand,  does 
it  require  more  or  less  heat  to  raise  a  given  weight  of  it  a  given  number  of  degrees  ?  Why? 
When  a  gas  can  expand  when  heated,  what  works  does  the  heat  imparted  perform  ?  When  it 
can  not  expand  ?  What  is  the  outer  work  in  the  latter  case  ?  Can  the  outer  work  ever  be  nega- 
tive ?    Give  a  familiar  example. 

When  we  abstract  heat  from  a  body  what  occurs  in  general  ?  Can  L  ever  be  positive  in  this 
case?    Give  an  example.    Is  TF ever  equal  to  zero  ?    Give  examples. 

Define  exactly  what  you  understand  by  "specific  heat."  Why  is  it  called  "specific?" 
What  is  the  specific  heat  of  water  ?  If  the  specific  heat  of  iron  is  i\; th,  what  does  that  mean  ? 
What  do  you  understand  by  "volume  capacity  "  for  heat  ?  What  by  "  weight  capacity  ?  "  How 
can  you  find  the  volume  capacity  from  the  weight  capacity  ?  What  is  real  specific  heat  ?  Appar- 
ent specific  heat  ? 

Wliat  is  the  disgregation  worli  in  solids  and  liquids  ?  How  does  it  compare  with  the  vibra- 
tion worlj  ?  Can  you  illustrate  this?  What  do  yon  understand  by  coefiicient  of  expansion? 
Why  does  the  specific  heat  increase  with  the  temperature  ?  Why  should  the  specific  heat  of  a 
body  when  liquid  be  greater  than  when  solid  ?  What  do  you  understand  by  specific  heat  for 
constant  volume  ?  What  for  constant  pressure  ?  Which  is  the  greatest  ?  Why  ?  When  we 
simply  say  "  specific  heat,"  without  further  limitation,  what  do  we  mean  ? 

What  do  you  mean  by  latent  heat  of  water  ?  Would  it  be  correct  to  say  latent  heat  of  ice  ? 
Why  not  ?  If  Co  is  the  specific  heat  for  constant  volume,  how  many  units  of  heat  would  bo 
necessary  to  raise  k  kilograms  of  a  body  t  degrees  ? 

How  many  pounds  of  mercury  at  the  temperature  of  300°  are  required  to  raise  15  pounds  of 
water  from  60°  to  70°  ? 

If  two  liquids  have  the  weights  w  and  w' ,  the  temperatures  t  and  I',  and  the  specific  heats  c 
and  c'  respectively,  what  is  the  temperature  of  the  mixture  ? 

Reduce  -  40°  Fahr.  to  Centigrade  degrees.  Reduce  -  273°  C.  to  Fahrenheit  degrees.  How 
do  you  reduce  generally  Fahrenheit  to  Centigrade  degrees,  and  vice  versa  ? 

What  outer  work  is  performed  when  2  pounds  of  air  are  heated  from  60°  to  70°  Fahr.  under 
the  pressure  of  the  atmosphere  ?    What,  when  3  kilogi-ams  are  heated  from  0°  to  1°  C.  ? 

What  is  the  specific  heat  of  air  under  constant  volume  ?  Under  constant  pressure  ?  Show 
how  to  find  from  these  the  mechanical  equivalent  in  French  measures  ?    In  English  measures  ? 

See  Examples  for  practice  at  end  of  volume. 


CHAPTEE  lY. 

EXPANSION   OF    GASES. — SPECIFIC    HEAT    OF    GASES. — DETERMINATION 
OP   MECHANICAL   EQUIVALENT   OP   HEAT. 


Expansion  of  Gases  iclien  Heated.— As  we  have  just  seen,  the 
disgregation  work  and  the  outer  work  are  very  small  indeed 
for  solid  and  liquid  bodies,  and  in  comjDarison  with  the  vibra- 
tion work  may  be  neglected.  But  for  gaseous  bodies  it  is  dif- 
ferent. Here  there  is  no  attraction  between  the  molecules,  or 
if  any,  it  is  exceedingly  small,  so  that  there  is  no  disgregation 
work.  All  the  heat  imparted  to  a  gas  goes,  therefore,  to  increase 
the  vibration  work,  that  is,  to  raise  the  temperature  and  to 
perform  outer  work.  This  last  is,  for  gases,  much  greater  than 
for  solid  and  liquid  bodies,  because  they  expand  much  more 
for  the  same  rise  of  temperature. 

Let  us  now  seek  to  ascertain  the  amount  of  this  expansion, 
as  well  as  the  other  properties  of  gases. 

Suppose  that  below  the  piston  EF,  Fig.  11,  in  the  cylinder 
ABCD,  we  have  one  cubic  meter  of  air  at  0^  and 
ordinary  tension,  corresponding  to  760"""-  of  the 
barometer.  This  air  weighs,  then,  according 
to  the  experiments  of  Eegnault,  1.29318  kilo- 
grams. If,  now,  we  heat  the  air,  it  expands  for 
every  degree  0I3  =  0.00367  of  its  volume.  This 
coefficient  has  been  determined  by  Eegnault 
and  Magnus  from  a  serie,s  of  very  careful  ex- 
periments. It  is  therefore  the  coefficient  of  ex- 
pansion of  air. 

If,  now,  this  air  volume  of  one  cubic  meter 
is  heated  to  2,  3,  4,  etc.,  degrees,  it  expands 

2  X  ^i^,  3  X  ^J-3,  4  X  3I3,  etc.,  or  2  x  0.00367, 

3  X  0.00367,  4  x  0.00367,  and  the  original  cu- 
bic meter  becomes,  at  these  several  tempera- 
tures. 


138  THEBMODTNAMICS. 

1  +  2  X  0.00367  cubic  meters, 
1  +  3x0.00367      " 
1  +  4x0.00367      " 

etc.     If  heated  to  f,  we  have,  therefore, 

1  +  0.00367^  cubic  meters. 

If  instead  of  one  cubic  meter  we  had  2,  3,  4,  or  in  general  V 
cubic  meters  to  start  with,  we  should  have  when  heated  to  t°, 

2  (1  +  0.003670, 

3  (1  +  0.003670, 
4(1+0.003670, 

or  generally 

F(l+ 0.003670  cubic  meters.    •     •    •    (HI.) 

If  heated  to  100"",  the  one  cubic  meter  becomes 

1  +  0.00367^  =  1  +  0.00367  x  100  =  1.367  cubic  meters, 
and  if  heated  273°,  we  have 

1  +  3I3  X  273  =  1  +  1  =  2  cubic  meters. 

If,  therefore,  air  is  heated  from  0°  to  273°,  it  expands  to 
double  its  original  volume. 

This  law,  according  to  which  air  expands,  is  called  the  law 
of  Gay-Lussac. 

It  is  evident  that  the  density  of  the  air  under  the  piston  EF, 
or  the  weight  of  a  unit  of  volume,  diminishes  as  the  volume 
increases.  Since,  for  example,  when  heated  273°,  our  one  cubic 
meter  becomes  two,  and  yet  still  weighs  1.29318  kilograms,  the 
density  at  this  temperature  is  only  one-half  of  that  at  0°. 

Since  the  densities  of  two  bodies  are  inversely  as  their  vol- 
umes, provided  that  the  weights  are  the  same,  we  can  find  the 
density  D,  of  the  air  for  any  temperature  t,  from  the  propor- 
tion, 

1 :  Z>  =  r(l  +  0.003670  :  V, 

where  we  assume  the  density  at  0°  =  1,  and  the  volume  at  this 
temperature  =  V. 

We  obtain,  therefore,  the  density  for  any  temperature  t, 

^"^  FOrT0.003670"'r+0:00367^'   "   *    ^^'^ 


EXPANSION  AND  SPECIFIC  HEAT  OF  GASES.  139 

1 


For  example,  for  t  —  10°  and  t  =  100°,  we  liave  D 


1  +  0.0367 


=  0.964:6,  om  D  —  z. —  0.7316.     One  cubic  meter  of  air 

1  +  U.Uob/ 

at  10°  weighs,  therefore, 

0.9646  X  1.29318, 
and  at  100°, 

0.7316  X  1.29318  kilograms. 

The  weight  of  one  cubic  meter  of  air  at  ordinary  tension 
(760™™  of  barometer)  and  t°  temperature,  is,  therefore, 

^  1.29318 

^       1  +  0.00367^  ' 

and  that  of  V  cubic  meters  is 

^  1.29318        ..,.,  ,^,, 

^  =  TToroo367f  ^^^^°^"^^'-  •  •  •  (^-^ 

Thus,  for  example,  3  cubic  meters  of  air,  at  ordinary  press- 
ure and  20°  temperature,  weigh 

^  3x1.29318  QfiifTTi 

^  =  1+0.00367x20  =  ^-^^^  kilograms. 

We  see  from  the  preceding  that  the  expansion  of  air  for  the 
same  rise  of  temperature  is  much  greater  than  for  solid  and 
liquid  bodies.  Experiments  have  also  shown  that  this  expan- 
sion is  very  nearly  the  same  for  all  gases.  Thus  Kegnault 
found  for  the  coefficient  of  expansion  of  hydrogen  0.003661,  and 
for  carbonic  acid  0.003710. 

It  seems  also  proved  by  experiment  that  the  coefficients  of 
expansion  of  such  gases  as  are  most  easily  liquefied  by  cold 
and  pressure,  as,  for  example,  carbonic  acid,  are  greater  than 
for  those  which  are  most  difficult  of  liquefaction.  We  accord- 
ingly assume  that  between  the  molecules  of  such  bodies  there 
is  a  certain,  though  small,  amount  of  attraction,  which  even  for 
the  so-called  "permanent"  gases  is  not  entirely  zero. 

Experiments  have  also  shown  that  the  coefficient  of  expan- 
sion increases  as  the  pressure  increases.  In  the  case  of  air, 
as  represented  in  the  figure  above,  the  piston  is  pressed  by 
the  atmosphere.     If,  however,  the  pressure  is  greater,  the  co- 


140 


THERMOD  TNAMIC8. 


efficient   of   expansion   increases   as    shown   in   the   following 
table  : 

Pressure  in  Coefficient 

Millimeters  of  of 

Barometer.  Expansion. 

^.^  j    760 0.0036706 

^^ (  2525 0.0036944 

XT  ^  (    760 0.0036613 

^^'^^^■^" {2520 ;...  0.0036616 

Carbonic  acid.,  i    "^'^ ^-^^^^f? 

(2545. 0.0038455 

[We  give  below  the  coefficients  of  linear,  surface,  and  cubic  expansion,  that  is, 
the  amount  by  which  a  piece  of  unit  length  area  or  volume  is  increased  for  a  rise 
of  temperature  of  one  degree  C,  counting  from  zero,  under  pressure  of  the  atmos- 
phere. 


I  Cubic  Expansion   Snrface Expansion  Linear  Expansion 
26.  6.' 


Lead 

Glass 

Platinum 

Gold 

Cast  Iron 

Copper 

Brass 

Silver 

Bar  Iron 

Steel  (untempered) 
Steel  (tempered) . . . 

Zinc 

Tin 

Mercury 

Water  * 


0.00008545 
0.00002584 
0.00002653 
0.00004398 
0.00003330 
0.00005155 
0.00005608 
0.00005726 
0.00003705 
0.00003236 
0.00003791 
0.00008825 
0.00005813 
0.00018018 
0.00046600 
0.003665 


0.00005697 
0.00001723 
0.00001768 
0.00002932 
0.00002220 
0.00003436 
0.00003-735 
0.00003817 
0.00002470 
0.00002158 
0.00002479 
0.00005883 
0.00003875 
0.00012012 
0.00031066 


0.00002848 
0.00000861 
0.00000884 
0.00001466 
0.00001110 
0.00001718 
0.00001868 
0.00001908 
0.00001235 
0.C0001079 
0.00001240 
0.00003942 
0.00001938 
0.00006008 
0.00015533 


TFuTTi 


8  TV  0  0 
92700 
TofoiJ 
l4Tl0Ti 
S-TBOO 


The  force  of  expansion  or  contraction  of  a  pi'ismatie  piece  for  a  change  of 
temperature  of  T  is 

P=8tFE 

where  8  is  the  coefficient  of  expansion,  E  the  coefficient  of  elasticity,  and  i^the 
area  of  cross-section.] 


Mechanical  Work  Performed  by  the  Air  during  Expansion. — Let 
us  now  compute  tlie  mechanical  work  which  one  cubic  meter 
of  air  performs  during  expansion. 

We  assume  that  the  piston  EF  has  an  area  of  one  square 
meter,  and  is  therefore  at  a  distance  of  one  meter  above  the 


'  Tlie  expansion  of  water  is  very  different  for  different  temperatures. 


EXPANSION  AND  8PE0IFIG  HEAT  OF  GASES.  141 

bottom  of  the  vessel  ABCD.  If,  tlien,  the  air  is  heated  273", 
the  piston  will  be  raised  one  meter,  and  the  air  pressure  will 
be  overcome  through  this  distance.  This  pressure,  at  0°  tem- 
perature and  760™™  height  of  barometer,  is  10334  kilograms  per 
square  meter  ;  so  that  the  work  performed  by  the  expanding 
air  is 

10334  X  1  =  10334  meter-kilograms. 

But  one  cubic  meter  of  air  at  0^  weighs  1.29318  kilograms, 
and  hence  the  work  performed  by  one  kilogram  of  air  under  the 
same  conditions  would  be 

1  9Q^i»  ~  7991.15  meter-kilograms. 

When,  therefore,  we  heat  one  cubic  meter  of  air,  free  to  ex- 
pand under  atmospheric  pressure,  from  0^  to  273°,  we  not  only 
increase  the  vibration  work,  but  we  also  obtain  an  outer  work 
of  not  less  than  10334  meter-kilograms.  We  see  that  the  outer 
work  in  the  case  of  solids  and  liquids  is  not  to  be  compared  to 
this. 

Heating  under  Constant  Volume. — If  we  conceive  the  piston 
EF  to  be  fixed,  so  that  it  cannot  be  raised  when  the  air  is 
heated,  then  evidently.no  outer  work  can  be  performed.  All 
the  heat  imparted,  therefore,  goes  to  increase  the  vibration 
work  or  to  raise  the  temperature.  With  the  temperature  the 
expansive  force  of  the  air  or  the  pressure  upon  the  piston  also 
increases,  and  becomes,  as  shown  by  experiment,  for  each  de- 
gree 2 Y^d  =  0.00367th  greater ;  that  is,  the  pressure  increases 
in  the  same  degree  as  the  volume  increased  in  the  first  case. 

If  we  denote  the  pressure  per  square  meter  at  0°  by^,  then 
the  pressure  for  a  rise  of  1  :  2,  3 1°  will  be 

P  +  2T3i^  =P(1+  2T3). 

_p  + 2.  313^=^(1+2^3). 
_p  +  3  .  ^\^p  =p{l+  gfs). 


pj^t.  j\-^p  =  j9  (1  +  2^3)  =pO-+  0.003670     •     .     (VI) 

For  t  =  273°,  the  pressure  is  evidently  2p),  or  twice  as  great 
as  for  0°. 

When,  therefore,  air  is  heated  under  constant  volume,  the 
expansive  force  increases  with  the  temperature. 


142  THEBMODTNAMICS. 

This  law  is  also  known  as  that  of  Gay-Lussac. 

It  may  be  explained  according  to  our  theoretical  views  as 
follows : 

When  heat  is  imparted  to  the  air,  the  velocity  of  the  atoms 
is  increased.  But  the  greater  this  velocity,  the  greater  the 
number  of  impacts  in  the  same  time  against  the  piston.  The 
greater  the  number  of  impacts  in  the  same  time,  the  greater 
the  pressure  of  the  expansive  force  of  the  gas. 

Absolute  Zero  of  Temperature. — Since,  now,  for  each  rise  of 
one  degree,  the  expansive  force,  or  what  is  the  same  thing,  the 
living  force  of  the  atoms  is  increased  o^sd  of  that  at  0^,  and  for 
every  fall  of  one  degree  is  diminished  gl^d ;  for  a  fall  of  273°, 
the  living  force  of  the  atoms  must  be  zero.  We  call  therefore 
the  temperature 

-  273°  C. 
the  "  absolute  zero."  * 

It  is  from  this  point  that  we  should  properly  reckon  the  ve- 
locity of  the  atoms  of  a  body,  or  its  living  force,  or  finally,  its 
temperature.  It  therefore  plays  an  important  part  in  the  me- 
chanical theory  of  heat.f 

*  [It  has  been  objected  to  this  reasoning  that  the  coefficient  of  expansion  af^d,  is  not  the  same 
for  all  gases,  that  it  varies,  especially  near  the  point  of  liquefaction,  and  that  it  also  depends 
upon  the  temptriUure  in  some  relation  not  yet  fully  known.  Thus  for  each  gas  there  is  a  dif- 
ferent absolute  zero,  and  nothing  justifies  the  assumption  of  this  special  one.  It  has  even  been 
termed  '"one  of  those  false  hypotheses  which  tend  to  retard  the  development  of  science." 

It  is  true  that  the  reasoning  above  seems  open  to  these  objections,  but  this  is  not  really  the 
reasoning  by  which  the  absolute  zero  is  properly  determined.  The  true  reasoning  cannot  be 
presented  in  an  elementary  manner  without  the  aid  of  the  highei-  mathematics.  It,  is  proved 
generally  by  the  principles  of  the  mechanical  theory  of  heat,  that  there  is  a  point  at  which  the 
living  force  of  the  atoms  would  be  zero,  and  that  this  pomt  must  be  the  same  f07'  all  bodies,  wh&ther 
there  is  di^^gl•egation  work  or  not.  In  some  cases  part  of  the  expansive  force  has  to  perform  more 
disgregation  work,  in  others  less.  In  the  text  we  have  simply  endeavored  to  simplify  the  deter- 
mination, and  to  illustrate  its  physical  significance  by  taking  a  body  in  which  the  disgregation 
work  is  nearly  zero,  and  thus  making  our  experiments  upon  a  gas  at  a  point  for  which  this 
assumption  is  known  to  be  approximately  correct.  Whether  this  seems  perfectly  correct  or  not, 
the  fact  remains  that  the  absolute  zero  is  a  point  which  has  a  definite  physical  significance,  and 
which  is  capable  of  more  or  less  accurate  determination,  for  all  bodies,  whether  perfect  gases  or 
not.  It  has  thus  been  found  to  be  very  closely  -  273°  C.  for  bodies  in  which  the  disgregation 
work  is  not  zero.  But  even  if  it  had  no  physical  significance,  which  it  has,  and  if  the  above  con- 
clusion were  founded  upon  the  corsideration  of  a  body  possessing  purely  hypothetical  properties, 
which  is  not  the  case,  still  it  would  not  follow  that  the  determination  of  such  a  point  would  be 
without  value.  The  coefficierd  of  elasticity  is  also  a  purely  supposititious  force,  which  will  stretch 
a  purely  hypothetical  body  by  its  own  length,  and  j'et  it  is  of  considerable  use  in  determinations 
of  strength  and  flexure  of  bodies,  and  can  scarcely  be  considered  as  "  retarding  the  development 
of  science." 

Let  it  be  remembered,  then,  that  there  is  an  absolute  zero,  and  that  it  is  the  same  for  ail 
bodies,  and  is  very  closely  -  273°  C,  as  determined  by  experiment.] 

+  This  corresponds  to  -459.°4  by  Fahrenheit's  scale. 


EXPANSION  AND    SPECIFIC  HEAT  OF  GASES.  143 

Since  in  the  last  experiment  no  outer  work  is  performed,  but 
all  the  heat  imparted  goes  to  increase  the  vibration  work,  we 
ought  to  expect  the  amount  of  heat  necessary  to  be  imparted 
for  a  certain  rise  of  temperature  to  be  less  than  in  the  first 
case,  where  outer  work  is  performed.  This  conclusion  is  con- 
firmed by  many  and  various  observations. 

[We  give  here  tables  of  the  most  remarkable  temperatures  : 

Greatest  artificial  cold —  140°  C. 

Mercury  freezes —  39.4° 

lee  melts 0° 

Greatest  density  of  water 4° 

Blood  heat 36.6° 

Water  boils 100° 

Red  heat 536° 

MELTING   POINT   OF   DIFFERENT   SUBSTANCES. 


C°. 

c°. 

Platinum 

2500 

1600 

1400 

1200 

1050 

1250 

1200 

1000 

900 

432 

360 

330 

260 

Tin.                             

230 

Wrought  Iron 

5  Tin,  1  Lead      

194 

Steel^ .....:..... 

8  Bismuth,  3  Tin,  5  Lead. . . . 
4  B.,1T.,1  Lead  (Rose's  metal) 

100 

Cast  Iron  (gray) 

94 

"      "    (white). 

109 

Gold  1100  to 

Yellow  Wax 

61 

Copper  1050  to 

Soda 

Potash 

90 

58 

49—43 

Paraffine 

46 

Zinc '. 

43 

Lead. 

Oil  Turpentine 

—  10 

Bismuth 

-39 

Water                      .... 

0 

BOILING   POINT   OF   DIFFERENT   SUBSTANCES. 


Mercury 

Sulphuric  Acid 

Sulphur 

Oil  Turpentine 


350 


440 
156 


Linseed  Oil 

Nitric  Acid 

Alcohol  (spec,  gr.  0.79  at  30'^) 

Sulphuric  Ether  

Sulphurous  Acid , 


316 

86 
78.4 
35 
-10] 


Calculation  of  tJie  Mechanical  Equivalent  of  Heat — When  the 
heated  air  is  free  to  expand  and  overcome  the  outer  air  press- 


144  THERMODYNAMICS. 

ure,  the  amount  of  lieat  necessary  to  be  imparted  to  one  kilo- 
gram, in  order  to  cause  a  rise  of  teinperature  of  one  degree  is 
0.23751  lieat  units.  This  is  the  "  specific  heat  "  of  air  for  constant 
pressure.  The  exact  determination  of  this  number  is  due  to 
Regnault. 

If  the  heated  air  cannot  expand,  if,  therefore,  the  volume  re- 
mains constant,  only  0.1684  heat  units  are  necessary  in  order 
to  raise  one  kilogram  one  degree.  This  is  the  "  specific  heat " 
for  constant  volume. 

The  specific  heat  of  the  air,  for  constant  pressure,  is,  therefore, 

0  23751 

A"TpoTr7  =  1-410  times  greater  than  that  for  constant  volume. 

This  number  has  been  determined  by  a  score  of  observations 
made  in  different  ways. 
The  excess  of  heat, 

0.23751  -  0.16847  =  0.06904 

heat  units,  is  that  which  goes  to  the  performance  of  outer 
work. 

We  have  already  seen  that  when  one  kilogram  of  air  is  heated 
from  0°  to  273°,  and  is  free  to  expand  under  the  air  pressure, 
the  outer  work  is  7991.15  meter-kilograms.     If  heated  from  0° 
to  1°,  then  the  outer  work  is 
7991.15 


273 


29.272  meter-kilograms. 


This  number  we  denote  generally  in  the  mechanical  heat- 
theory  by  the  letter  i^.*  A  work  of  29.272  meter-kilograms 
corresponds  to  an  expenditure  of  heat  of  0.06904  heat  units. 
One  unit  of  heat,  then,  corresponds  to 

A  AAQni.  ~  423.98  meter-kilograms. 

This  result  agrees  perfectly  with  that  found  by  Joule  as  the 
mean  of  a  large  number  of  experiments.  We  can  now  easily 
deduce  a  general  formula  for  the  mechanical  equivalent. 

If  we  denote  the  specific  heat  of  air  for  constant  volume,  that 
is,  the  number  0.1687,  by  c„,  and  that  for  constant  pressure,  or 
0.23751,  by  Cp,  then  the  difference  c-,  —  c„  denotes  the  amount 

*  [Vov  Fahrenheit  degrees  and  foot  lbs.  E  =  53.354.  The  student  will  do  well  to  make  the 
calculation.] 


MECHANICAL  EQUIVALENT  OF  HEAT.  145 

of  lieat  corresponding  to  the  outer  work  of  29.272  meter-kilo- 
grams, or  to  the  work  R.     For  one  unit  of  heat,  therefore,  we 

obtain  the  mechanical  work   (mechanical  equivalent)  —: ,  from 

the  proportion 

Cp-c,  :1  ::  B  ij^. 

Hence  we  have  -;  = . 

A      Cp-c, 

Since  Cj,  =  1.410^,,  we  have,  when  we  substitute  for  1.41  the 
letter  Jc, 

1=      ^ 

-ifX  ^y"^  Cy 

or,  since  it  is  customary  to  write  (7„  without  the  index, 

i  =  ^h)- <™-) 

If  we  assume  the  older  determination  of  the  specific  heat  for 
constant  pressure  of  Delaroche  and  Berard,  of  0.267,  we  obtain 
a  smaller  value  for  the  mechanical  equivalent.  It  is  thus  that 
Mayer  found  (1842)  the  number  365,  Holtzmann  (1845)  374, 
and  Clausius  (1850)  370. 

[The  value  of  i?  in  French  measures  and  Centigrade  degrees  is,  as  we  haye 
seen,  for  air  29.273  meter-kilograms.  We  give  here  the  values  of  B  for  other 
gases  both  in  French  measures  and  in  English  measures  for  both  Centigrade  and 
Fahrenheit  degrees. 

French  Measures.  English  Measures. 

Centigrade.  Cent.  Fahr. 

B.  R. 

Air 29.273  meter-kil. .. .     96.0376  53.354  foot  lbs. 

Nitrogen 30.134        "  ....     98.867  54.926 

Oxygen 26.475        "  ....     86.863  48.257 

Hydrogen 433.613        "  ....1386.579  70.333 

These  can  all  be  calculated  from  the  formula 

Tf  _(^p  —  <^v  _  P^  _   P 

A        ~    T  ~  yT' 

where  y  is  the  density,  or  weight  of  one  cubic  unit  of  volume. 

The  value  of  —  is  424  meter-kilograms  for  French  measures  and  Centigrade 

degrees,  1,390  foot  lbs.  for  English  measures  and  Centigrade  degrees,  and  773 
foot-pounds  for  English  measures  and  Fahrenheit  degrees. 
10 


146 


THERMOD  TNAMIGS. 


We  give  below  the  values  of  Cp  and  cv. 

Cp 


Air  0.16847 

Nitrogen 0.17273 

Oxygen 0.15507 

Hydrogen 3.41226 


k: 

1.4098 
1.4114 
1.4026 
1.4132 


Cp. 


Weight  of        Weight  of 
Cubic  Meter  1  Cubic.  Foot 
n  Kilograms,     iu  Pounds. 


0.23751. . .  .1.29318. . .  .0.08073 
0.24380. . .  .1.25616. . . .0.07860 
0.21751....  1.42980. ...0.08936 
3.40900. . .  .0.08957. . .  .0.00559] 


Increase  of  tJie  Expansive  Force  by  Compression.  —  Mariotte's 
Law. — When  we  lieat  the  air  without  allowing  it  to  expand,  its 
expansive  force  increases  at  the  cost  of  the  heat  imparted,  but 
its  density  remains  the  same.  Let  us  now  increase  the  expan- 
sive force  in  another  way,  viz.,  by  compressing  the  air.  Accord- 
ing to  our  principles  the  action  is  evident ;  for  by  compressing 
the  air  we  perform  mechanical  work,  and  this  can  always  re- 
place a  certain  amount  of  heat. 

If,  then,  we  assume  below  the  piston  EF,  Fig.  12,  one  cubic 
meter  of  air  at  0^  temperature  and  atmospheric  pressure,  the 
expansive  force  will  increase  as  the  piston  is 
forced  down.  If,  for  example,  the  air  is  com- 
pressed into  half  its  former  volume,  its  tension  is 
twice  as  great ;  if  compressed  to  one-fourth  of  its 
former  volume,  its  tension  is  four  times  as  great ; 
and  so  on.  In  such  case  we  assume,  indeed,  tliat 
the  temperature  is  kept  constant,  viz.,  at  0^.  In  the 
same  degree  in  which  the  tension  increases,  the 
density  evidently  increases  also. 

This  law,  according  to  which,  for  constant  tem- 
perature, the  tension  of  a  gas  increases  as  its 
volume  decreases,  is  called  Mariotte's  laiv. 

This  law  also  is  a  necessary  consequence  of 
our  assumptions  as  to  the  constitution  of  gases. 

Every  atom  in  the  cubic  meter  of  air  makes  at  0°  a  certain 
number  of  impacts  upon  the  piston,  and  thus  causes  a  certain 
pressure.  If,  at  the  same  temperature,  the  gas  only  occupies 
half  its  original  volume,  the  atom  makes  in  the  same  time 
double  as  many  impacts,  because  its  velocity  (temperature)  is 
the  same,  and  it  has  only  half  as  far  to  go.  Its  pressure  upon 
the  piston  is  therefore  twice  as  great,  and  since  this  is  the 
case  with  all  the  atoms,  the  tension  of  the  entire  mass  must  be 
twice  as  great. 


EXPANSION  AND  8PEGIFIG  HEAT  OF  GASES.  147 

If  we  denote  tlie  Yolume  of  air  by  V^  and  its  tension  by  p^ 
(upon  tlie  square  meter),  then  cliange  tlie  volume  Fi  into  K, 
and  the  pressure  jJi  into  j)^,  we  have,  according  to  Mariotte's  law, 

Fi :  ^2  : :  p. :  pi,  or 
V^p,  =  V,p,. 
If  the  volume  V^  with  the  tension  p.  is  changed  to  V3  with 
the  new  tension  jh,  we  have, 

F2  :  Fg  : :  ^3  :  P-i,  or 
V^Pi  =  Vaps- 
Comparing  with  the  above,  we  have 

V,p,=  V,p,  =  nB,eic (YIII.) 

We  see,  therefore,  that  the  product  of  the  volume  and  press- 
ure is  constant,  ijrovided  that  the  temperature  is  kept  the  same. 

As  now  the  expansive  force  increases,  the  density  also  in- 
creases, and  inversely. 

If  the  density  for  the  volume  Fi  is  Di,  and  for  /^  =  D«,  we 
have 

FiiF^::  A:A,  or 
V,D,  =  KD,. 
On  the  other  hand, 

Z>i '. D.2  \  \ px'.  P2. 

The  density  increases,  therefore,  inversely  as  the  volume,  and 
directly  as  the  tension. 

Example  1. —  What  is  the  weight  of  ^th  cubic  meter  of  air  at  0°  and  4  atmos- 
pheres 9 

One  cubic  meter  at  0°  and  atmospheric  pressure  weighs  1.39318  kilograms, 
hence  ^th  of  a  cubic  meter  weighs  0.16165  kilograms.     We  have  then 

1:4::  0.16165  :  x. 
Hence 

a;  =  4  X  0.16165  =  0.6465  kilograms. 

Example  2. — What  is  the  volume  of  one  Tcilogram  of  air  at  0°  and  \th  atmos- 
phere 9 

One  kilogram  of  air  at  0°  and  at  atmospheric  pressure  has  a  volume  of 

^   f-oo-iQ  =  0-'7733  cubic  meters.     Its  volume,  therefore,  at  a  tension  of  ^th  of  an 
atmosphere,  is  given  by 

0.7733  :  a;  :  :  i  :  1,  or 
a;  =  5  X  0.7733  =  3.8665  cubic  meters. 


148  THEBMODYNAMICS. 

Ilariotte's  and  Gay-Lussac's  Laivs  Combined. — If  we  denote 
the  volume  at  0°  and  ordinary  tension  by  Vq,  tlien  for  j^ressures 
of  2,  3,  4  Pi  atmospheres,  the  volume  will  be,  according  to 
Mariotte's  law, 

n    v^  K  n 

2  '    3  '   4  ' Pi' 

If  the  temperature  of  the  air  is  t^,  it  is  evident  that  for  the 
same  tension  p^  its  volume  will  be  greater.  Since  for  each 
degree  the  air  expands  0.00367  of  its  volume,  the  volume  Vq 
will  become  at  ti°  Vo  (1  +  0.00367^i),  and  we  have 

'  Pi 

Again,  if  the  tension  at  0°  were  p^,  the  volume  would  be 

at  0°,  and  if  the  temperature  were  ^2°  instead  of  0°,  the  volume 
would  be 

Tz  -.  To  (1+0.003674) 


have, 

then. 

''2  - 

P^ 

Vi   ; 

:  r,  :: 

^{1 
Pl 

+  0.00367^  :  ^  (1 

+  0.003674 

Vi 

Pi 

1  +  0.00367^1 
1  +  0.003674 ' 

.  .  . 

(IX.) 

The  law  expressed  by  this  formula  is  known  in  physics  as 
the  combined  law  of  Mariotte  and  Gay-Lussac. 

Since,  further,  the  volumes  are  inversely  as  the  densities, 

A_^     1  +  0.00367^1  ,-yr. 

Di" Pi'  1  +  0.00367^2 ^    '' 

Example  1. — A  quantity  of  air  0/  F,  =  1  cuMe  meter,  4  =  10%  and  Pi—l 
atmosphere,  is  compressed  to  V^  =0.8  cubic  meter,  and  t.2  =  100°.  What  is  the 
tension  ? 

We  have  from  IX,, 

J__^     1  +  0.00367x10  1-03^7 

0.8"  1    '1  +  0.00367x100'  ^-^^ -i's   1367' 

1.25x1.367      1.70875      ,  „,„    ,  , 

^'^  ^^  =  -i:0367"  =  T:0367  =  ^'^^^  atmospheres. 


EXPANSION  AND  SPECIFIC  HEAT  OF  GASES.  I49 

Example  2. — If  a  mass  of  air  of  V^  =  30  cubic  meters,  ^i  =  1  atmosphere, 
and  t-^  =  10°,  in  passing  through  the  hloiving  apparatus  of  a  Mast  furnace  is 
heated  to  ifj  =  200°  and  compressed  ^0  1.26  atmospheres,  what  ivill  ie  its  new 
volume  ? 

We  have  from  IX., 

^_L36      1  +  0.00367  x  10 
V^~     1     *  1  +  0.00367  X  200'  ^^ 
30  X  1.7340 


1.26  X  1.0367 


39.8  cubic  meters. 


Example  3. — If  the  density  of  air  for  p^  =  760"™-  height  of  barometer,  and 
t^  =  0°  is  1,  what  would  he  the  density  for  p^  —  750™"-  and  to  =  20°  ? 
We  have  from  X., 

^  _  750       1  +  0.00367  x  0 
1    ~  760  ■  1  +  0.00367  x  20  ' 

^^  ==  S  •  O^l  =  sTfesl  =  ^•^^^^' 

and  hence  the  weight  of  one  cubic  meter  in  the  new  condition  would  be  only 
1.29318  x  0.9195  =  1.189  kilograms. 

Transformation  of  the  last  tivo  Formulae. — In  the  meclianical 
theory  of  lieat  these  last  formulae  are  put  into  a  simpler  form. 
First  we  denote  the  coefficient  of  expansion  by  a,  and  can  thus 
write 

V^      Pi       1  +  at.^  '  D^      2h      1  +  at^' 

If  now  we  divide  numerator  and  denominator  of  the  right 
side  of  these  equations  by  a,  we  have 

1       .  1       . 

^  =  ^.f ,    and    ^~  =  Pl.- . 

a  a 

Since  now  a  —  giy,  and  hence  -  =  273,  we  have 

V^  _p^     273  +  t^  ^_i'^     273  +  t^ 

F/^p/ 273  +  ^3  Z)i  ~^/ 273  +  ^3* 

From  the  first  of  these  equations  we  have 

V.p.     _     V-2P. 
273  +  4~273  +  ti 

In  like  manner  we  have  for  the  volume  V3,  the  tension  p^,  and 
the  temperature  4, 

JFiP^_  -  _  ^■'■P' 

273"+  ^i~273  +  ts* 


150  THERMO  D  YNA  MIC8. 

We  have,  therefore,  generally, 

273  +  t,      273  +  h~  273  +  ^3'  '     *     ^      ' 

It  thus  appears  that  the  above  quotients  have  a  constant 
value  for  any  "  perfect "  gas.* 

The  formulae  hold  good,  evidently,  when  V■^  and  V,.  are  the 
"specific  volumes,"  that  is,  the  volumes  of  the  unit  of  weight, 
as  the  kilogram. 

If  therefore  Vq  is  the  specific  volume  for  i^  —  0°  and  atmos- 
pheric pressure  ip^,  and  v^  that  for  the  temperature  tx  and  ten- 
sion j9i,  we  have 

273  +  ^0      273  +  ^1' 
But  now  the  volume  Vq  for  the  pressure  j)o  and  temperature 
^0  is  known.     For  since  1.29318  kilograms,  under  these  circum- 
stances, is  known  to  occupy  the  space  of  one  cubic  meter,  one 
kilogram  will  occupy 

L29318  '^^^^  ^^*^^'- 

This  is  therefore  the  specific  volume  Vq  at  0°. 
If,  now,  we  express  p^  in  kilograms  per  square  meter,  or  put 
po  =  10334  kilograms,  we  have 
1 


vopo         1.29318  ""  -^^^^^  10334  10334 


:=  29.272. 


273  +  to  273  +  0     1.29318  x  273  ~  353.03814 

We  have  then 

273n:-273i:^-'^^-^^^- 

This  number  we  have  already  found  to  be  the  outer  work 
performed  under  atmospheric  pressure  by  one  kilogram  of  air 
when  its  temperature  is  raised  from  0°  to  1°.  This,  as  already 
remarked,  we  denote  by  R,  and  have  thus 

^_       V,p,       _       V,p,       _       V2P2  .^ 

273  +  to~  273  +  t,~~  273  +  L'        ' 
where  Vq,  Vi,  v^,  are  the  specific  volumes,  or  volumes  of  one  kilo- 

*■  [Meaning  by  "  perfect  gas  "  one  between  whose  molecules  there  are  no  forces  of  attraction, 
or  one  so  far  removed  from  its  point  of  liquefaction  that  the  disgregation  work  may  be  dis- 
regarded.] 


EXPANSION  AND  SPECIFIC  HEAT  OF  OASES.  151 

gram  of  the  gas,  and  p^,  p^,  p^,  are  tlie  specific  pressures,  or 
pressures  upon  the  square  meter. 

lip  is  expressed  in  atmospheres,  we  have 

For  other  gases,  which  have  different  densities,  B  has  differ- 
ent values. 

We  have  already  remarked  that  the  absolute  zero  of  temper- 
ature lies  at  273^  below  the  zero  of  the  centigrade  scale.  At 
this  temperature  the  living  force  of  the  atoms  is  0,  and  the 
body  possesses  no  heat. 

From  this  point  the  heat  of  a  body,  or  its  inner  work,  should 
be  measured.  If,  then,  a  body  has  a  temj)erature  of  0°  accord- 
ing to  the  thermometer,  its  actual  temperature  is  273°,  and  so 
for  the  temperatures  t^  ti,  U,  etc,  the  absolute  temperatures 
are  273  +  U,  273  +  h,  273  +  ^3,  etc. 

Thus  we  see  that  the  denominators  in  the  equations  above 
give  the  absolute  temperatures.  If,  then,  we  denote  these  by 
jTi,  T2,  etc.,  we  have 

^~   To-   T^  -    T, 

That  is,  if  ive  divide  the  specific  volumes,  multiplied  by  their  cor- 
responding pressures,  by  the  corresponding  absolute  temperatures, 
the  quotients  are  constant  and  equal  to  B,  or  for  air  to  29.272. 
From  these  equations  we  obtain 

RTo  --=  vopo  ) 

BT,  =  v,p,  V (XII.) 

BT,  -.^  v,p,  ) 
etc. 
In  this  form  the  combined  laws  of  Mariotte  and  Gay-Lussac 
can  be  easily  remembered.     We  shall  have  occasion  to  make 
frequent  use  of  these  formulae.* 

Example  1. — What  is  the  volume  v-,  of  one  kilogram  of  air  at  the  tempera- 
ture t^  =  100°  and  pressure  p-^^  =  5000  kilograms  p)6r  square  meter  ? 
We  have  29.373  (373  +  100)  =  v^x  5000, 

,  39.373x373      10918.456       „  ,^^^      ,. 

hence         v^  = — =  — ~-  ^ —  =  2.1887  cubic  meters. 

*  [If  V  is  the  volume  of  G  kilograms  of  ga?,  then  —  is  the  volume  of  one  kilogram,  oi  v  —  ~, 
and  hence  GBT=  T>.] 


152  THERMODYNAMICS. 

Therefore,  2,  3,  4  .  ,  .  kilograms  of  air  would  occupy  the  space  of  2  x  2.1837, 
3x2.1837,  4x2.1837  cubic  meters,  etc. 

Example  2. — If  one  Mlogram  of  air  has  a  volume  of  3  cuhic  meters  and  tem- 
perature of  200",  what  is  its  pressure  per  square  meter  9 

We  have  29.272  (273  +  200)  =  8^^,  or_p,  =4615.218  kilograms. 
Example  3. — What  is  the  temperature  t^  of  one  kilogram  of  air  whose  ten- 
sion isp.^  —  3  atmospheres  and  volume  v,  =0.5  cubic  meters  9 
We  have  0.002833  (273  +  ^,)  =  0-5  x  3, 
or  0.7734  +  0.002833^,  =1.5 
0.002833^1  =  0.7266 
^1  =  256.4  degrees. 


QUESTIONS   FOE    EXAMINATION. 

What  is  the  disgiegation  work  in  a  perfect  gas  ?  What  effects  are  produced  hy  heat  im- 
parted ?  How  does  the  outer  work  compare  with  that  for  solid  and  liquid  bodies  ?  What  is  the 
coefficient  of  expansion  for  air  ?  If  F cubic  meters  of  air  are  heated  i°  under  atmospheric  press- 
ure, what  is  the  new  volume  ?  What  is  the  law  of  expansion  of  air  called  ?  State  it  concisely 
in  words.  How  does  the  density  vary  with  the  temperature  for  constant  pressure  ?  How  can  you 
find  the  weight  of  V  cubic  meters  of  air  at  t°  ?  Does  the  coefficient  of  expansion  vary  for  differ- 
ent gases  ?  HoAV  ?  For  what  gases  is  it  least ';  Does  it  vary  with  the  pressure  ?  How  ?  What 
is  the  cause  of  variation  ?  What  is  a  "perfect  gas?"  Would  it  varj'  for  such  a  gas  ?  If  we 
heat  one  cubic  meter  of  air  under  atmospheric  pressure  from  0°  to  273°  C,  what  is  the  outer  work 
performed  ?  What  would  it  be  for  one  kilogram  of  air  ?  What  is  the  disgregation  work  ? 
The  vibration  work  ?    The  inner  work  ? 

When  air  is  heated  under  constant  volume,  what  effects  does  the  heat  produce  ?  How  does 
the  pressure  increase  with  the  temperature  ?  If  the  pressure  at  0°  is  ii,  what  is  it  at  1°  ?  At  2°  ? 
At  i°  ?  What  is  this  law  called  ?  State  it  in  words.  How  can  you  explain  it  according  to 
theoretical  views  ? 

What  do  you  understand  by  "  absolute  zero  ?  "  Has  this  point  a  definite  physical  signif- 
icance ?  Is  it  the  same  for  all  bodies  ?  If  it  hiid  no  physical  significance,  would  it  necessarily 
follow  that  it  is  valueless  in  the  theory  ?  Which  is  the  greater,  the  specific  heat  for  constant 
pressure  or  for  constant  volume  ?  Why  ?  What  is  the  ratio  of  the  two  for  air  ?  What  does  R 
denote  in  our  notation  ?    Deduce  a  relation  between  the  two  specific  heats  R  and  A . 

State  concisely  Mariotte's  law.  Under  what  assumption  as  to  temperature  does  this  law  hold 
good  ?  Is  it  a  consequence  of  our  theoretical  views  as  to  the  constitution  of  gases  ?  How  ? 
ir  P]  and  »,  are  initial  pressure  and  volume,  what  relation  subsists  between  them  and  any  other 
Tp  and  «  ?  Is  this  law  exact  for  all  gases  ?  For  any  gas  ?  What  do  we  call  a  gas  for  which  it  is 
exact?  What  constitutes  a  "perfect"  gas?  Are  there  any  such?  Wliat  relation  subsists 
between  volume  and  density  ?  Between  pressure  and  density  ?  State  again  Gay-Lussac's  laws. 
State  algebraically  the  combined  laws  of  Mariotte  and  Gay-Lussac.  Deduce  a  relation  which 
must  exist,  by  virtue  of  these  laws,  between  the  volumes,  pressures,  and  temperatures  at  two 
different  states,  and  the  coefficient  of  expansion.  Between  the  densities,  pressures,  and  tem- 
peratures. Deduce  from  these  last  the  simplest  expression  of  the  combined  laws.  Does  this 
relation  between  volume,  pressure,  and  temperature  hold  good  for  all  perfect  gases  ?  What  does 
R  signify  ?    What  is  it  for  air  ? 

What  is  the  volume  of  8  kilograms  of  air  when  its  pressure  is  5000  kilograms  per  square 
meter  and  temperature  100°  C.  ? 

If  one  kilogram  of  air  has  a  volume  of  3  cubic  meters  and  a  temperature  of  100°,  what  press- 
ure must  it  have  ? 


EXPANSION  AND  SPECIFIC  HEAT  OF  GASES.  153 

If  2  kilograms  of  air  have  a  tension  of  3  atmospheres  and  a  vohime  of  1  cubic  meter,  what 
must  be  the  temperature  ? 

How  many  kilograms  per  square  meter  correspond  to  one  atmosphere  ? 

If  8  cubic  feet  of  air  are  heated  from  0°  to  8°  C,  what  is  the  new  volume  ?  From  0°  to  S° 
Fahr.  ?    What  is  the  density  in  each  case  ? 

What  IS  the  weight  of  3  cubic  feet  of  air  at  atmospheric  pressure  and  25°  C.  temperature  ? 
At  25°  Fahrenheit  ? 

If  2  cubic  feet  of  water  are  heated  under  atmospheric  pressure  from  0°  to  100°  C,  what 
expenditure  of  work  is  equivalent  to  the  heat  imparted  ?    From  40°  to  212°  Fahrenheit  ? 

What  is  the  coefficient  of  expansion  for  air  ?  If  one  cubic  foot  of  air  is  heated  from  0°  C.  to 
273°  C,  what  is  the  new  volume  ?  If  it  is  further  heated  from  273°  to  274",  what  is  the  increase 
of  volume  ?  Is  this  Increase  -ihad  of  the  volume  at  273°,  or  of  the  volume  at  0°  ?  If  10  cubic 
feet  of  air  are  heated  from  11°  C.  to  12°  C,  what  is  the  increase  of  volume  ?  Is  this  increase  or-jd 
of  the  10  cubic  feet,  or  2  7  :sd  of  what  the  volume  of  the  10  cubic  feet  tvoidci  be  at  0°  ?  Has 
pressure  any  influence  upon  the  coefficient  of  expansion  ?  What  influence  ?  Does  the  coefficient 
of  expansion  vary  for  different  gases  ?  Define,  then,  exactly  what  is  meant  by  coefficient  of  ex- 
pansion ? 


CHAPTEE  V. 

HEAT  CURVES  AND  THE  MECHANICAL  WORK  WHICH  A  GAS  PERFORMS 
DURING  EXPANSION  AND  RECEIVES  DURING  COMPRESSION. 

Isothermal  Curve. — Let  OX  and  0  T  be  two  lines  perpendicu- 
lar to  eacli  otlier,  tlie  so-called  "  co-ordinate  axes,"  OX  being  the 
axis  of  abscissas,  and  0  Y  the  axis  of  ordinates. 

Suppose  that  we 
have  in  a  cylinder  the 
unit  of  weight  of  air 
(one  kilogram)  of  the 
volume  OB  =  V,  and 
tension  BF  =  p,  and 
the  temperature  t  If 
this  air  expands  to 
double  its  volume,  or 
to  0G=20B  =  2v, 
and  if  we  suppose  that 
during  the  expansion  the 
temperatMre  t  is  con- 
stant, that  is,  that  heat 
is  imparted  to  the  air 
from  without  as  its 
temperature  falls  during  expansion,  then  the  tension  becomes 
CH=^BF  =  ip.  If  the  volume  becomes  0I)  =  30B  =  Sv, 
the  tension  becomes  1)1=  \BF  =  \p,  etc.  If  we  join  the  points 
FHIK,  the  curve  gives  the  law  of  variation  of  the  tension  with 
the  volume.  We  call  the  curve  thus  obtained  the  "  isothermal 
curve  for  permanent  *  gas."  It  represents  graphically  the  law 
of  Mariotte. 


*  [Late  investigations  have  shown  thatall  the  so-called  "  permanent"  gases  can  be  liquefied. 
The  term  is  therefore  to  be  taken  merely  as  applying  to  those  gases  which,  under  ordinary 
circumstances,  are  so  far  from  their  point  of  liquefaction  that  the  disgregation  work  in  expan- 
sion can  be  neglected.] 

154 


ISOTHERMAL  CURVE. 


155 


llecJianical  Work  ivJiich  the  Air  performs  during  Expansion  and 
receives  during  Compression. — In  order  to  find  the  meclianical 
work  whicli  tlie  air  performs  during  its  expansion,  we  must 
suppose  EE  divided  into  an  indefinitely  large  number  of  parts, 
that  is,  we  must  suppose  tlie  volume  v  to  increase  little  by  lit- 
tle, and  find  the  corresponding  tensions.  The  pressure  during 
the  small  increase  of  volume  may  be  regarded  as  constant,  and 
we  thus  obtain  the  work  during  this  increase  by  multiplying 
the  pressure  by  the  change  of  volume.  The  sum  of  the  pro- 
ducts thus  obtained  gives  the  mechanical  work  during  expan- 
sion. 

[Such  a  summation  can  easily  be  made  by  means  of  the  cal- 
culus. 

Thus  if  the  initial  tension  is  i\  and  volume  t\,  and  if  after 
expansion  we  have  p)  and  v,  we  have 

Pi  :  p  '.  :  V  :  Vi,         whence        p  —  ^-^—^ . 

During  the  small  expansion  dv,  the  pressure  p  may  be  regarded 
as  constant,  and  the  work  performed  is  therefore 

,        Pi  Vi  dv 
pdv  = . 


Integrating  this  between  the  limits  v  and  Vi,  we  have  the 
work 


L—p-i^Vi  log  nat -, 


or  in  common  logarithms 


2.3026  7)  V  loi 


^1 


i:  =  2.3026  vi^i 


log  — 


where  v  is  the  greater  volume  and  v^  the  less. 
Since  p^  :  p  :  :  v  :  v^,  ^e  have  also 


L  =  2.3026  lov  log 


p- 


(XIII.) 


(XIV.) 


where  pi  is  the  greater  tension  and  jo  the  less.] 

The  same  formulae  hold  good  when  the  specific  volume  v  and 
tension  p)  are  by  compression  under  constant  temperature 
changed  into  the  less  volume  v^  and  greater  tension  p^.     They 


156  THERMODYNAMICS. 

evidently  give  tlie  work  per  unit  of  area  of  the  piston  or  sur- 
face pressed  upon,  since  p  is  the  pressure  per  unit  of  area. 

Example  1. — What  work  must  be  expended  in  order  to  convert  0.33.cubic  meters 
.  of  air  at  ordinary  tension  (760™™-)  iiito  air  of  814mm.  tension,  the  temperature 
being  kept  constant  ? 

If  we  denote  the  original  volume  by  V,  and  tension  hj  p,  and  the  new  greater 
tension  by^^,  we  have 

L  =  2.3026  Vp  log  ^, 
^     ^   p' 

or  substituting  the  values  given, 

L  =  2.3026  X  0.33  x  1.0334  log  Hn  =  233.99  meter-kilograms. 

Example  2. —  What  is  the  mechanical  work  which  the  unit  of  weight  ofadr 
performs  when  it  expands  to  double  its  volume,  the  temperature  being  constant,  its 
volume  V,  beiyig  772-9-3-nj-  =  0.7733  cubic  meters,  p^  =  1.0334  kilograms  per  square 
meter,  and  v  =  2vi  ? 

We  have  L  =  2.3026  p^v,  log  —  ,  or 

L  =  2.3026  X  0.7733  +  1.0334  log  2  =  5538.6  meter-kilograms. 

Example  3. — The  piston  of  a  steam  engine  has  an  area  of  O.lJj.  square  meters. 
We  have  beneath  it  a  volume  of  steam  0.395  meter  high  and  3  atmospheres  tension. 
What  mechanical  work  is  performed  when  the  piston  moves  0.658  meters,  the  tem- 
perature remaining  constant  9 

The  original  volume  Fj  is  0.14  x  0.395  cubic  meters.  The  volume  F  after 
expansion  is  0.14  (0.395  +  0.658).     Hence 

V      0.395  +  0.658      1.053 


V,~         0.395         ~  0.395  ■ 


We  have  then 


L  =  2.3026  X  0.14  x  0.395  x  3  x  10334   log  J^  =  1679  meter-kilograms. 

The  work  of  the  steam  during  the  full  pressure  is  0.14  x  10334  x  3  x  0.395  = 
1714  meter-kilograms. 

Hence  the  total  work  performed  is  1679  +  1714  =  3393  meter-kilograms.  If 
this  is  performed  in  one  second,  we  have  a  work  of  m^  =  45.2  horse-power 
(French). 

Amount  of  Heat  imparted  or  abstracted  during  Expansion  or  Com- 
pression, according  to  Mariotte's  Law. — We  know  from  what  has 
preceded  that  when  a  gas  expands  while  performing  work  its 
temperature  must  sink,  because  the  outer  work  is  performed  at 
the  expense  of  the  inner. 

If,  therefore,  during  the  expansion  the  temperature  remains 
constant,  heat  must  be  imparted  from  without. 


I80DYWAMIC  CURVE.  157 

Since,  now,  the  temperature  or  vibration  work  remains  the 
same,  the  outer  work  performed  is  the  exact  equivalent  of  the 
heat  imparted.  If  therefore  we  denote  the  heat  imparted 
during  expansion,  measured  in  heat  units,  by  Q,  we  must  have 

424  Q  =  2.3026  vp  log  -. 

Since   we   denote  the  mechanical  equivalent  of  heat,  424,  by 
■^ ,  we  have  generally 

■^  =  2.3026  i^i)  log  -, 

or  Q  =  2.3026  Avp  log  - (XV.)     . 

Just  as  during  the  expansion  of  a  gas  we  must  impart  heat 
in  order  to  preserve  the  temperature  constant,  we  must  abstract 
heat  during  compression.  The  work  performed  upon  the  gas 
goes  to  increase  its  vibration  work,  or  its  sensible  temperature. 
The  heat  abstracted  must  therefore  be  equal  to  the  work  per- 
formed upon  the  gas.     We  have  thus,  in  this  case  also, 

Q  =  2M2Q  Avp  log-. 

Isodynamic  Curve.* — For  solid  and  liquid  bodies  the  case  is 
not  so  simple.  "When  such  a  body  is  compressed,  that  is,  when 
outer  work  is  performed  upon  it,  we  cannot  directly  determine 
how  much  of  this  outer  work  goes  to  increase  the  vibration 
work,  and  how  much  to  disgregation  work.  It  may  be  that  one 
or  the  other  of  these  parts  is  zero  or  negative,  as  we  have 
already  seen. 

We  have  therefore  for  solid  and  liquid  bodies  the  isodynamic 
curv%  which  gives  the  relation  between  pressure  and  volume 
when  the  inner  luork  (that  is,  both  the  vibration  work  and  the 
disgregation  work)  is  constant. 

Since  in  gases  there  is  very  little,  if  any,  mutual  action  be- 
tween the  molecules,  and  therefore  no  work  is  required  to  bring 
them  nearer,  the  outer  work  performed  upon  the   gas   only 

*  [Sometimes  called  also  "  isenergic  curve,"  or  curve  of  equal  energy.  When  there  is  no  dis- 
p:regatioii  work  the  isothermal  curve  corresponds  to  the  isenergic  or  isodynamic] 


158  THEBM0DYNAMIC8. 

increases  tlie  vibration  work.     Here  tlien  tlie  isodynamic  curve 
becomes   identical   with   the   isothermal.     All  that   has   been 
said  with  reference  to  the  isothermal  curve /or  gases  holds  good 
therefore  for  the  isodynamic  curve  also. 
If  in  the  formula 

Q  =  2.3026  Avp  log  - , 

or  Q  =  2.3026  Av^jh  log  -  , 

we  put  in  place  oipv  or  p^Vi  the  values  from  Equation  XIL,  viz., 
RT  and  RT^,  we  have 


Q  =  2.3026  ART  log   - 


(XVI.) 


^  =  2.3026  ^i^Ti  log-^ 

where,  as  before,  v  is  the  greater  volume  and  pi  the  greater 
pressure. 

From  these  formulae  we  can  determine  Q  when  the  initial 
and  final  volumes  or  pressures  and  the  temperature  are  known. 
Since  pv  =  piVi,  so  also  RT  =  RT^,  or  the  temperature  is  con- 
stant, as  should  be. 

EXAMPLE. 

We  have  one  kilogram  of  air  inclosed  in  a  cylinder.  The  temperature  is  ^  = 
30\  What  work  will  it  perform  when  it  expands  from  the  less  volume  v^  to  the 
greater  v  =  ^v^,  and  how  many  units  of  heat  must  be  imparted  to  keep  the  tem- 
perature constant? 

We  have  for  the  work 

L  =  2.3026  BT^  log  ^  =  2.3026  ET^  log  f. 
Or  by  substituting  the  numerical  values 

L  =  2.3026  X  29.272  (373  +  30)  x  0.125  =  2549  meter-kilograms. 

Since  now  Z  "^  ^       ^^        ^~  ^^'  ^^  ^^^^  ^'^^  ^ 

4^4  X  2549  =  6.012  heat  units. 

Adiabatic  Curve.^—li  a  gas  expands,  all  the  time  performing 
work,  without  any  heat  being  imparted  to  it  from  ivithout,  the  outer 
work  which  it  performs  can  only  be  at  the  expense  of  the  vibra- 

*  [Sometimes  called  also  "  ise?)fro]nc  curve,"  or  curve  of  equal  "  entropy  ; '"  entropy  being 
defined  as  that  property  of  a  body  that  remains  constant  when  the  body  undergoes  any  change, 
bat  without  receiving  or  losing  any  heat.] 


ABIABATIC  CURVE.  159 

tion  work  or  temperature.  The  temperature  then  diminishes 
as  the  expansive  force  diminishes.  This  last  then  diminishes 
for  two  reasons ;  by  reason  of  the  increase  of  the  volume,  and 
by  reason  of  the  decrease  of  temperature. 

We  see  at  once  that  the  work  of  the  gas  for  the  same  expan- 
sion must  be  less  than  when  the  temperature  is  kept  constant 
by  imparting  heat  from  without. 

If  Vi  is  the  specific  volume  and  p^  the  tension  of  air,  and  v 
and  p  that  after  expansion,  we  have  now  no  longer,  according  to 
Mariotte's  law, 

V  :Vi  ::p^  '.p,        or        j9y  =  p^v-^, 

but  we  have  the  relation,  first  proved  by  Poisson,* 

*  The  above  relation  was  proved  by  Laplace  and  by  Poisson  upon  the  hypothesis 
of  the  caloric  theory  of  heat.  It  is  easily  deduced  from  the  mechanical  theory  of 
heat  by  the  aid  of  the  calculus.  For  those  who  wish  to  understand  the  method 
of  deduction  we  give  it  hei'e.  Others  must  accept  it  simply  as  an  accurate  ex- 
pression of  the  law  of  relation  of  pressure  and  volume  during  adiabatie  change. 

We  have  for  every  perfect  gas 

pv  =  BT, 
or  pdv  +  vdp  —  RdT, 

hence  c?T=  ^^^^ (1.)  ' 

If  we  denote  inner  work  by  i[7and  outer  by  L,  then 
Q  =  A{U^L), 
or  dQ  =  A{dU+dL). 

For  adiabatie  change,  dQ  =  0,  and  hence 

dU+dL  =  0 (2.) 

Now  for  a  perfect  gas  there  is  no  disgregation  work,  and  ^C/"  represents  vibra- 
tion work  or  change  of  temperature  only.     Hence 

du^-;^dT. 

A 
Also,  since  the  pressure  for  a  very  small  change  of  volume  may  be  considered 
as  constant, 

dL  =  pdv. 

Substituting  in  (2)  and  referring  to  (1)  we  have 

j^  {pdv  +  vdp)  +  pdv  =  0 
for  the  differential  equation  of  the  adiabatie  curve. 
But  from  Equation  VII.  we  have  -j-^  =  TZr\'  ^^^^^e 

r — z  {pdv  +  vdp)  +  pdv  =  0, 


160  THEBMODYNAMIGS. 

or  ^v^-"  =  ^iVi^ ", 

(XVII.) 


P 

where  v  is  the  greater  volume  and  pi  the  greater  pressure. 
The  law  is  very  similar,  as  we  see,  to  Mariotte's,  only  the  vol- 
umes are  raised  to  the  power  denoted  by  1.41,  or  in  general  by 

— ,  i.  e.,  the  ratio  of  the  specific  heat  by  constant  pressure  to 

the  specific  heat  by  constant  volume.  This  law  has  therefore 
been  very  appropriately  called  by  Eedtenbacher  the  expo- 
nential laio  of  Mariotte,  and  as  such  it  may  easily  be  remem- 
bered. 

If  we  make  use  of  logarithms,  we  have 

1.41  log  -  =  log  ^ . 

°  Vi  ^   p 

If  we  denote  the  ratio  1.41  by  h,  we  have 

^  log  -  =  log  ^. 

If  we  assume  the  initial  volume  v,  =  1,  and  the  initial  ten- 
sion pi_  ~  \  atmosphere,  we  have  for  the  tension  p  when  the 
air  has  expanded  to  double  its  volume,  ox  v  =  ^v^,  at  the  ex- 
pense of  its  inherent  heat,  that  is,  without  any  heat  being 
imparted  from  without, 

1.41  log  2  =  log  — , 

or  pdv  +  vclp  +  kpdv  —  pdv  =  0, 

or  hpdv  +  vdp  =  0. 

Dividing  hj  pv,  we  obtain 

dp      Tcdv      . 
—  H =  0. 

p         ■v 

Integrating  this  betewen  the  limits  of  the  initial  pressure  and  volume  (^i  and 
Vi),  and  the  final  pressure  and  volume  (p  and  v),  we  have 

log^  —  logj5i  =  A;  log  w,  —Jo  log  V, 
log^=/fcloglS        or         P=r-l-)\ 
pv^  —p^v^^  —  etc. 
For  air  the  ratio  of  the  specific  heats  k  =  1.41,  hence  we  have 


IDIABATIC  CURVE. 


161 


1.41  X  0.3010 


log  — 


0.42441  =  log 


0.  376  atmosp. 


,      or      U.4Zi4ti±   =  102'  — 

p  ^  p 

or  —  =  2.6671,     or     p  =  ^rw^r-r^ 

p  ^        2.6571 

If  tlie  air  had.  expanded  according  to  Mariotte's  law,  that  is, 

if  heat  had  been  imparted  from  without  in  such  a  manner  that 

while  expanding  the  temperature  remained  constant,  we  should 

have  had 

p  =  0.5  atmospheres. 

It  is  now  easy  to  calculate  in  similar  manner  the  tension  p, 
which  the  air  has,  when,  without  receiving  heat  from  without, 
it  expands  to  3,  4,  etc.  times  its  original  volume.  If  we  should 
thus  actually  compute  these  tensions,  and  lay  them  off  as 
ordinate  s,  with  the 
corresponding  vol- 
umes as  abscissas, 
the  curve  BODE 
thus  obtained  would 
give  the  relation  of 
volume  to  pressure. 
This  curve  is  called 
the  adiabatic  curve 
for  permanent  gases. 
We  see  that  it  ap- 
proaches the  axis  of 
abscissas  much  more 
rapidly  than  the 
isothermal  or  isody- 
namic  curve,  which 
is  represented  by 
the  broken  line.  Let 
us  now  determine 
the  decrease  of  temperature  of  the  air  during  the  expansion. 

We  have  found  by  Equation  XII.,  for  the  combined  laws  of 
Mariotte  and  Gay-Lussac, 

pv  =:   BT, 

piVi  =  BT^,  etc., 

where  T  and  Ti  are  the  absolute  temperatures  at  the  volumes 
11 


162  THERMODYNAMICS. 

V  and  Vi.     From  Equation  XVlI.  we  have  for  acliabatio  expan- 
sion 

—  \     =  J^      or       --—  =  =^ 


If  we  multiply  both  sides  by  v-^  and  divide  by  v,  we  have 


vvi  pv  \ViJ  pv 


or  generally,  putting  li,  for  the  ratio 


or  generally  Kj)     ^  -^ (XYim) 

B.T        T 

The  right  side  of  this  equation  becomes  nrj^=  -7^,  and  hence 

vy-_  T,        273  +  t, 

Cv  ' 

We  can  therefore  find  the  final  temperature  t,  from  the  initial 
temperature  U  and  the  expansion  .ratio  —  . 

We  see,  then,  from  the  formula,  that  the  0.41  poivers  of  the  spe- 
cific volumes  are  inversely  proportional  to  the  corresponding  absolute 
temperatures. 


EXAMPLE. 

If  the  specific  volume  Vj  of  the  air  has  the  temperature  t^  =  30',  what  will  be 
its  temperature  t  when  it  has  expanded  to  v  =  2vi,  performing  work  and  ivithout 
receiving  heat  from  without  ? 

We  have        /a^.V"",  ^^3  +  30        ^r         20.^1-^^ 
We  have        (^  ^  J      -    373  +  ^  ^^         ^      -   273  +  T 

or 
1.3286  (273  +  0  =  308     or     362.7078  +  1.3286^  =  303.         1.3286^  =  -  59.7078. 
Hence  t  ~  44.9°. 

fv  \"-"       T 
Since  in  the  formula  ( —  j     =  ~^,  the  expansive  force  p  or 

Px  does  not  occur,  the  end  temperature  depends  solely  upon 
the   initial  temperature   and  the   expansion  ratio.      Whether 


ADIABATIG  CURVE.  163 

therefore  the  specific  volume  is  small  or  great,  and  hence  the 
expansive  force  great  or  small,  makes  no  difference  in  the  final 
temperature,  if  only  the  initial  temperature  remains  the  same. 

Just  as  we  have  determined  the  final  temperature  from  the 
initial  and  end  volumes,  so  we  can  also  determine  it  from  the 
initial  and  final  tensions. 

Thus  we  have  for  the  law  of  adiabatic  expansion, 


Q- 

-  Il 
P 

This  can  be  written 

XXa...i^  =  i^fi'  = 

1                   rplAX 

.  /^N-    or     ^  . 

\p  )                       V, 

iff- 

.  (XX&.) 

If  we  multiply  the  numerators  by  p,  and  the  denominators  by 
p„  we  have 

vp  _  ppr">'' 
v,pi     pip'-'"'' ' 
This  gives  us 

vp_  ^  fp\ 

PiV^        \p,  J 

k-i 

T 

Hence                  ^  = 

,^.0..90T           273   +  ^ 

'■\p-j    -  m  +  t. 

.     .     . 

(XXIa.) 

or  generally 

& 

.     .     . 

(XXI6.) 

Hence,  the  0.2907  poivers  of 

the  pressures  are  directly  propor- 

tional  to  the  absolute  temperatures. 


EXAMPLE. 
We  have  in  a  cylinder  one  unit  in  weight  of  air,  at  a  tension  j)i  =  1^  atmos- 
pheres, and  a  temperature  of  t^  —  30°.     What  will  be  its  temperature  t  when  the 
air  has  expanded  adiabatically  until  its  tension  j?  is  only  one  atmosphere  ? 
We  have 

'3\o.2907      273  +  ^ 


M' 

).2907 

278 

+  t 

h) 

278 

+  30 

273 

+  t 

or 

273 

.-.,  ...-,--  ^    ,  303 

Hence 

+  t  =  269.306       or       t=-^ 

or  in  round  numbers  t  =  —  3.7°. 

The  temperature  therefore  falls  33.7°. 


164  THEBM0DTNAMIG8. 

V  /  73  \  0.1093 

From  Equation  XXa  —  =  /  i-i  J 

we  can  now  find  at  once  the  expansion  ratio,  or  the  ratio  of  the  final  Tolume  to 
the  original. 

^  /  3  \  0.7093 

We  thus  have  —  =  ( —   )         ; 

Vi       V  2  / 

hence  -  =  1.333         or         -  =  ^. 

While,  then,  the  tension  falls  in  the  ratio  of  3  to  2,  the  volume  increases  in  the 
ratio  of  3  to  4. 

Outer  Work  Performed  hy  Air  when  expanding  Adiabatically. — 
The  question  now  arises,  What  loork  does  the  air  perform  when 
it  expands,  performing  work,  without  receiving  heat  from 
without  ? 

Since  the  expansion  occurs  at  the  expense  of  the  vibration 
work,  or  of  the  temperature,  the  work  performed  must  depend 
upon  the  initial  and  final  temperatures. 

Now  we  know  that  under  constant  volume  we  must  impart 
0.16847  heat  units,  in  order  to  raise  the  temperature  of  one 
weight  unit  of  air  one  degree.  This  heat  we  have  called  the 
specific  heat  of  air  for  constant  volume.  In  like  manner  we 
must  abstract  0.16847  heat  units  from  each  kilogram  for  every 
degree  that  we  cool  it,  under  constant  volume.  But  0.16847 
heat  units  correspond  to  a  mechanical  work  of 

0.16847  X  424  meter-kilograms, 

and  this  work  must  be  performed  when  the  unit  weight  of  air 
is  cooled  one  degree  by  expansion,  while  performing  work,  be- 
cause the  work  performed  is  the  equivalent  of  the  heat  which 
disappears,  since  no  heat  is  imparted  or  abstracted  during  ex- 
pansion. 

We  denote  the  specific  heat  for  constant  volume,  or,  in  the 
case  of  air,  the  number  0.16847  by  c,  and  the  mechanical  equiv- 
alent of  heat  (424  meter-kilograms)  we  denote  by  -j,  and  hence 
the  work  is 

cx^  =  -J  meter-kilograms. 

If,  therefore,  the  specific  air  volume  has  the   temperature 


ABIABATIG  EXPANSION— OUTER   WOBK.  165 

ti,  and  lience  tlie  absolute  temperature  273  +  fi,  the  inlierent  vi- 
bration work  in  it,  or  the  "  intrinsic  energy,"  as  it  is  called,  is 

2Ci  —  -r  {2i7d  +  ti)  meter-kilograms. 

If  now  this  volume  gradually  expands,  overcoming  an  outer 
pressure  which  at  any  moment  is  less  than  the  air  pressure  by 
an  infinitely  small  amount,  until  its  temperature  is  t,  so  that  t 
is  less  than  fi,  then  the  inner  work  inherent  in  it  will  be 

u  =  ^  (273  +  t)  kilograms. 

The  inner  work  which  disappears  is  thus 

u^  -u  =  ^  (273  +  ti)-^  (273  +  0 

=  ^  ^(273  +  f,)  -  (273  +t))^  ^  {f^  -  f)  meter-kilograms, 

and  this  is  evidently  exactly  equal  to  the  outer  work  performed, 
since  no  heat  has  been  imparted  from  without.  If  we  denote 
this  outer  work  by  L,  we  have 

L  =  ^{f,-f),    .     .     .     .     (XXIIa.) 
or  also, 

L  =  ~{T,-  T)  .     .     .     .     (XXIK.) 
=  71.431  (^1  —  t)  meter-kilograms.* 

Thus  we  see  that  the  outer  work  is  jwoportional  to  the  difference 
betiveen  the  initial  and  filial  temperatures,  as  might  have  been  at 
once  concluded. 

The  area  BCDEFG  (see  last  Figure)  inclosed  by  the  adia- 
batic  curve  BCBE  and  the  ordinates  EG  and  EF,  represents 
the  work  performed  during  expansion,  or  the  decrease  in  the 
inherent  vibration  work. 

The  final  temperature  t  in  the  last  formula  can  be  found  from 
Equation  XlXa, 

273  +  ^1 


V  J 


273  +^  ' 


*  [This  is  the  worli  performed  by  one  unit  of  weight.  For  G  units  of  weight  we  h;ive 
71.431  G  (i!,  —  t).  If  wo  use  Fahrenheit  degrees  and  English  measures  we  have  130.00 (?  (^  -  0- 
If  we  use  Centigrade  degrees  and  English  measures  we  have  234  173 (r  (^,  -  t).  The  student  will 
do  well  to  make  the  reductions.] 


166  THERMODYNAMICS. 

or  from  Equation  XXIa, 

"'      273  +  t 


\pj  273  +  t, ' 

If  Q  is  tlie  number  of  disappearing  lieat  units,  we  liave 

L  =  —r  Q,  and  lience 

L  =  ^  Q  =  ^{t.-t), 

ox  Q  =  c{ti  —  t)  heat  units. 

If  L  or  the  outer  work  is  equal  to  zero,  that  is,  if  the  air  per- 
forms no  outer  work  during  expansion — if  there  is  no  outer 
pressure  overcome — we  have 

0=  ^{t,-t)     or    0  =  tr-t, 

OT   t  =  fi. 

That  is,  the  final  temperature  is  equal  to  the  initial  temperature,  and 
the  temperature  of  the  air  remains  unchanged,  as  the  experiment  of 
Joule,  already  noticed,  clearly  proves. 

EXAMPLE. 

What  is  the  work  performed  by  the  air  in  the  last  example,  when  the  tem- 
perature sinks  from  +  30°  to  —  3.7'  ? 

Here  t,  -^  =  38.7°,  hence 
L  =  71.431  X  33.7  =  2407.225  meter-kilograms. 
The  number  of  heat  units  disappearing  is 

Q  =  0.16847  X  33.7  =  5.6774. 

We  may  also  determine  the  work  L  directly  from  the  initial 
and  final  volumes,  or  from  the  initial  and  final  tensions,  as  well 
as  from  the  initial  and  final  temperatures. 

Thus  we  have 

L=^  (273  +  ^0  -  -^-  (273  +  0- 

If  we  divide  by  273  +  ^i,  we  have 

L       _  ^  _  ^     273  4-^ 
273  +  ^i~^       a'  273  +  fi' 

The  factor  on  the  right  is,  according  to  Equation  XXIa= 


ADIABATIC  EXPANSION. 


167 


and  since  R  (273  +  ti)  =  v^p^,  or  273  +  t, 


—,  we  have 


Vipx      A       A   \p 
or 

Again,  since  according  to  Equation  XlXa,  :^r^ —  =  (  — 

we  have  also 


Compression  of  Air  ivlienHeat  is  neitlier  Imparted  nor  Abstracted. 
— When  air  is  compressed,  the  opposite  phenomena  take  place. 
The  work  expended  in  the  compression  is  transformed  into  heat. 
The  vibration  work  is  there- 
fore increased.  The  tension 
of  the  air  is  then  increased, 
for  two  reasons.  First,  the 
density  is  increased  by  the 
compression,  and  the  atoms 
strike  oftener  against  the  pis- 
ton which  causes  the  compres- 
sion, and  this  alone  causes  an 
increase  of  tension.  Second, 
the  living  force,  or  the  veloc- 
ity of  the  particles,  is  in- 
creased by  the  heat  due  to 
the  transformation  of  the 
work,  and  this  also  causes 
an  increase  in  the  expansive 
force.  This  last  must  in- 
crease according  to  the  same 
law  as  before,  during  expan- 
sion, it  diminished. 

That  air  is  heated  by  com- 
pression has  long  been  known.     We  are  all  familiar  with  the 
"  pneumatic  syringe."     This  consists  of  a  glass  or  metal  cylin- 


IQQ  THERMODYNAMICS. 

der,  in  whicli  moves  an  air-tight  piston.  Upon  tlie  under  side 
of  tlie  piston  is  a  piece  of  tinder.  If  now  the  piston  is  pressed 
quickly  down,  the  air  is  comjpressed,  heat  is  developed,  and 
the  tinder  ignited. 

If  we  denote  the  specific  volume  of  air  inclosed  in  a  cylinder 
by  Vi  and  the  pressure  by  ^Ji,  and  if  v  and  p  are  the  volume 
and  pressure  after  compression,  we  have  from  the  law  already 
found,  when  heat  is  not  imparted  nor  abstracted, 

\Vx)  p' 

Let  now.  Fig.  15,  OB  =  Vi  =  1  be  the  initial  volume,  and  pi  =  1 
atmophere  be  the  corresponding  pressure.  If  we  compress 
the  air  to  J  of  its  original  volume,  we  have 

^'^  P  ' 

or  3">  ^  4> "  ; 

hence  p  —  (|)^''  =  1.5  atmos.  =  DE. 

For  V  =  -|-  =  OF,  we  have 

at""  = -,  hence»  =  2^", 

p  ^ 

or  }:>  =  2.657  atmospheres  =  FG. 

liv  =  \=  OH,  we  have 

(i)' "  =  - )  hence 
p  =  4' "  =  7.06  atmospheres  =  HI. 
The  curve  joining  the  points  CEGI  thus  found  gives  the  law 
of  the  increase  of  pressure  as  the  volume  diminishes.     We  see 
how  rapidly  this  curve  rises  for  great  diminution  of  volume. 

The  area  BGEGHI,  inclosed  by  the  curve  and  the  ordinates 
BG  and  HI,  gives  the  mechanical  work  expended  in  compress- 
ing the  body,  and  which  is  therefore  completely  transformed 
into  heat. 

Example  1. — In  a  cylinder  we  have  one  kilogram  of  air  of  0°  and  atmos- 
pheric pressure.  What  amount  must  it  be  compressed  adiabatically  in  order 
to  raise  the  pressure  to  2  atmospheres  ;  how  much  is  the  air  heated  ;  what  work 
is  necessary  for  compressing  it;  and  how  many  heat  units  appear? 

We  have  from  Equation  XXa, 


P 
hence  v  =  O.G1188w, 


AD  IAEA  TIG  EXPANSION.  169 

Since  f,  is  the  speeiiie  volume,  or  volume  of  unit  of  weight,  at  0',  which 
volume  is  .^      =  0.773  cubic  meters,  we  have  for  the  volume  of  v 

V  =  0.773  X  0.61188  =  0,4731  cubic  meters. 
The  final  temperatui-e  t  is  given  by  Equation  XXIa, 

373  +  t 


.2907 


373+0  • 

Hence  3"-«'0'  x  373  =  373  +  ^  =  334.01, 

or  t  =  334.01  -  373  =  61.01  \ 

The  air  therefore  becomes  heated  to  61.01'.  The  mechanical  work  necessary 
for  the  compression  is,  from  Equation  XXII., 

L  =  ^{i,-t)  =  71.431  5  (0  -  61.01)  =  -  4358  meter-kilograms. 

The  negative  sign  denotes  that  this  work  is  received  by  or  performed  upon 
the  gas,  instead  of  performed  by  it. 
The  quantity  of  heat  generated  is 

Q  =  AL=  ~,f^^  =  -  10.38  heat  units. 
434 

If  we  compress  only  i  or  :^  of  a  kilogram  to  3  atmospheres,  only  ^  or  ^  as 

much  work  would  be  necessary,  and  we  should  generate  only  ^  or  ^  as  many  heat 

imits.     But  the  rise  of  temperature  would  be  just  the  same,  as,  evidently,  this 

depends  only  upon  the  expansion  ratio. 

Example  2. — If  a  caloric  engine  compresses  adiabatically  at  each  stroke  1  of  a 

kilogram  of  air  at  10°  and  1  atmosphere  to  4  atmospheres,  what  mechanical  work 

is  necessary,  and  how  much  is  the  air  heated  ? 

We  have  from  Equation  XXIa, 

^Y'-'""'    _  373  +  ^  /4 y-^^"  _   373  +  ^ 

^pj  ""373  +  /,    ^^    \lj  "373+10' 

hence  /  ^  _  273  +  383  x  40--'9»'  =  -  373  +  433.62  =150.63% 

The  mechanical  work  is 

C  71  431 

i  =  i -^  (/,- 0  =^  ^^^f^  (10  -  150.63) 

=  17.86  X  -  140.63  =-  3511.47  meter-kilograms. 

If  this  work  is  performed  in  one  second,  we  have 

3511.47       „„  ,„  , 
— 7— —  =  33.48  horse-power. 
75 

If  the  air  thus  compressed  is  confined  and  heated  373%  its  pressure  would  be 
doubled,  or  8  atmospheres.  The  temperature  after  heating  is  then  373  +  150  63 
=  433.63% 

What  mechanical  work  can  the  air  now  perform,  when  it  expands  adiabeti- 
cally,  until  its  tension  becomes  again  one  atmosphere  and  temperature  at  10°  ? 

W^e  have  L  =  \j{t^-t)  =  {x  71.431  (433.63  -  10) 

=  17.86  X  413.63  =  7387.35  meter-kilograms, 

or  if  the  work  is  performed  in  one  second, 

7387.35       „„  ^  . 

1^ —  =  98.5  horse-power. 

75 


170  THEBMOD  YNAMICS. 

The  compression  of  the  air  consumes  thus  about  \  of  the  work  which  the 
heated  air  can  perform.  If  we  assume  that  the  resistances  consume  about  40  per 
cent.,  we  have  the  total  loss  of  work,  40  +  33^  =  78;f  per  cent.  Only  about  26 j 
per  cent,  remain  as  useful  work.  In  most  cases  the  efficiency  would  probably  be 
lower. 

[As  we  often  have  occasion  to  make  use  of  the  f ormulse  for  adiabatic  expansion 
or  compression,  the  following  table  will  be  found  useful  in  abridging  calculations. 

The  table  gives  for  different  values  of  the  ratio  -^  the  corresponding  ratios  of 

the  temperatures,  volumes,  etc.  It  is  made  out  for  compression,  or  p^  greater 
than  jj  I .  It  applies  equally  well  to  expansion  if  we  simply  suppose  all  the  sub- 
scripts at  head  of  columns  interchanged.  Before  giving  the  table,  we  group 
below,  for  convenience  of  reference,  the  adiabatic  formula  already  deduced. 


For  air, 


A;  =  1.41,     A;-l  =  0.41,      ^-,    =2.44,      ^=0.7098, 
7c-l 


0.2907,     T ^  =  3.44. 


Work  done. 


~  AB 
In  these  formulae,  for  air 


^.,^.,[i-^]=^.,i,,[i-^]. 


0.16847,         -i^=2.44, 
AB 


and 


1  c 

-r  =  424,       -  =  71.431,       B  =  29.272  in  French  measures  and  Centigrade 

A  A 

degrees. 

1  r 

-J  =  1390,       J  =  234.1733,      B  =96.0376  in  English  measures  and  Centi- 
grade degrees. 

4  =  772,       -^  =  130.0588,       B  =  53.354  in  English  measures  and  Fahren- 
A  A    .  ■  ° 

heit  degrees. 
T=t  +  273  Centigrade  =  t  +  459.4  Fahrenheit. 
1  atmosphere  =  10334  kil.  per  sq.  meter  =  14.7  lbs.  per  sq.  foot. 


ADIABATIG  CHANGE. 


171 


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163  tOMOSC 


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—  -    ■       - 70T<J  O  tOO»  iOOCOOD 


172  THEBM0DTNAMIC8. 

We  can  illustrate  the  use  of  this  table  by  the  examples  already  given.     Thus, 

Example  1. — In  a  cylinder  we  have  one  kilogram  of  air  of  0'  and  atmospheric 
pressure.  What  amount  must  it  be  compressed  adiabatically  in  order  to  raise 
the  pressure  to  two  atmospheres  ?  how  much  is  the  air  heated  ? 

Here  —  =  2.      Opposite  2  in  the  table  we  find  at  once—  =  0.6117,  hence 

Vo  =  0.6117  vi,  the  same  as  already  found  by  calculation. 

We  have  also  at  once  from  the  table,  corresponding  to  —   =   2,  ~  =  1.2226; 

hence  T,  =  1.2226  Ti,  or  t,  +  273  =  1.2226  x  273;  hence  t,  =  333.77  -  273  = 
60.77^ 

Example  2. — If  a  calorie  engine  compresses  adiabatically  at  each  stroke  |-  of 
a  kilogram  of  air  at  10°  and  1  atmosphere,  to  4  atmospheres,  how  much  is  the 
air  heated  ? 

Here  —  =  4,  and  we  have  at  once  from  the  table 
Pi 

^   =  1.4948,     or    t,  +  273  =  1.4948  x  283,     or  t,  =  150^ 

V-2 

The  new  volume  is  —  =  0.3741,     or    v-2  =  0.3741 1\. 

Example  3. — If  {-  kilogram  of  air  at  4  atmospheres  tension  and  150'  expands 
adiabatically,  performing  work,  till  its  tension  is  1  atmosphere,  what  is  its  new 
temperature  ? 

For  expansion  we  simply  have  to  invert  the  subscripts  at  the  head  of  the  col- 
umns.    The  student  will  do  well  to  make  a  special  note  of  this.     Considering, 

Pi  T-^ 

then,  all  the  subscripts  reversed,  we  have  from  table  for  i—  =  4,     Tjf  =  0.6690; 

Pi  Tx 

hence  T.,  =  0.6690^1  =  0.6690  x  423,     or    t.  +  273  =  282.98,     or    ti  =9.98°,  or 

about  10°,  as  should  be. 

In  similar  manner  we  can  easily  find  pressure  and  temperatures  when  volume 
ra,tio  is  given,  or  pressure  and  volume  when  temperature  ratio  is  given.  The 
student  will  do  well  to  propose  other  examples  and  solve  them  both  by  table  and 
calculation.     (See  examples  at  end  of  Part  I.)    The  use  of  the  last  two  columns 

for    J  will  be  explained  hereafter.     They  have  no  reference  to  temperature  at 

all.] 

Transference  of  Air  from  one  Adiahatic  Curve  to  Another. — 
We  have  now  to  deduce  a  very  important  tliermodynamic 
principle,  of  whicli  we  shall  have  occasion  to  make  frequent 
use. 

Suppose  the  specific  volume  Vi,  with  the  tension  p^,  and  the 


ADIABATIC  TRANSFER  OF  AIR. 


173 


absolute  temperature  T^,  to  be  compressed,  without  any  heat 
being  added  to  it  from  without  or  abstracted  from  it.  The 
compression  thus  takes  place  according  to  the  adiabatic  curve, 
or,  as  we  say,  this 
curve  represents  the 
law  of  the  change  of 
condition  of  the  air. 
If  the  volume  Ov^,  = 
Vi  is  compressed  to 
Ov.2  =  v-j,  let  p.2  and 
Ti  be  the  corre- 
sponding tension 
and  absolute  tem- 
perature. If  the 
volume  is  Qv^  =  v^, 
let  p:i  and  T-i  be  the 
corresponding  ten- 
sion and  absolute 
temperature. 

Now  suppose  that 
the  air  with  the  vol- 
ume Vi,  the  pressure 
Pi,  and  the  absolute 
temperature  T^,  or, 
as  we  may  say  briefly,  "in  the  condition"  Vip^Ti,  has  heat 
imparted  to  it  in  such  a  manner  that,  while  it  expands  to  the 
volume  10^,  and  the  tension  sinks  to  q^,  the  temperature  T^  re- 
mains constant. 

Then  let  the  volume  lUi  be  compressed  adiabatically,  so  that 
the  curve  BB  represents  the  relation  between  the  volume  and 
tension  at  any  instant. 

During  the  passage  of  the  air  from  the  condition  v^p^T^  to 
the  condition  w^q^Ti  (while  therefore  the  tension  changes  ac- 
cording to  the  isothermal  or  isodynamic  curve  T^T-^),  let  the 
quantity  of  heat  imparted  be  Q-^. 

Suppose,  again,  the  air  in  the  condition  v,p.,To  to  expand, 
heat  is  imparted  to  it  in  such  a  manner  that  the  temjoera- 
ture  T^  is  preserved  constant,  until  it  arrives  at  the  condition 
iv,q2  T.  in  the  adiabatic  curve  BB,  and  let  the  heat  imparted 
be  Q:. 


174  THERMODYNAMICS. 

Tlien  we  shall  prove  this  relation  : 

or, 

the  lieat  imparted  in  tJie  first  case  is  to  that  imparted  in  the 

as  the  corresponding  temperatures  T^  and  To,  at  which  the  addition 

of  heat  commenced,. 

If,  again,  the  air  passes  from  the  condition  v,pi T-^  to  the  con- 
dition iv^jiiTz,  and  if  Q  is  the  heat  imparted,  then  we  shall 
have 

Hi    '•     V3    •    •    -^  2    •    -^3* 

"We  have  then  generally, 

Q,  :  Q.,  :Q^=T,'.T,'.  T,. 
Let  us  now  seek  to  prove  this  relation. 
The  law  of  the  adiabatic  curve  AA  is  from  Equation  XXIa 

.  ,0.2907  rjl 

or  it  is  also 

,^.    0.2937   _       rp^ 


In  like  manner  the  law  of  the  adiabatic  curve  BB  is  given  by 

^^-%- (^■) 

and 

^    V    0.2907  m 


(■ 


„/  r, (^- 


The  amount  of  heat  Qi  which  is  necessary  to  change  the  gas 
from  the  condition  v-^p-,  T^  to  the  condition  w^qx  jTi  ,  along  the 
isothermal,  is  from  Equation  XVI. 

a  =  2.3026^7^^1  log  ^. 
In  like  manner 

(^2  =  2.3026  J^r,  log  ^. 
and 

a  -  2.3026  ^i^^a  log  ^. 

We  have  therefore 

Q.  :  Q.  :  a  =  ^.log^  :  ^2log^  :  ^alog^. 
q\  2'2  <iz 


EXPANSION  OF  AIB.  I75 

From  Equations  (a)  and  (c)  we  liave 

P^^^     or     2l^1^ 

Px      gi  gi      q^' 

In  like  manner  from  (h)  and  (c/), 
We  have  then  generally 

Pi     _    P-2     _     Ps 

qi   ~  q-2  "  qi' 
Hence  in  tlie  above  proportion  tlie  logarithms  of  the  quotients 

P}_      P^     £3^   are  equal,  and  we  obtain 
^1       q^       q-s 

Qr.  Q-r.  Q,=  T,:Tr.T„ 
which  was  to  be  proved. 

If  tlierefore  we  wish  to  change  the  condition  of  air,  as  determined 
hy  a  certain  point  on  an  adiahatic  curve,  into  another  condition  tvhich 
lies  on  anotJier  adiahatic  curve,  hy  the  addition  of  heat,  under  con- 
stant temperature,  the  quantity  of  heat  which  must  he  imparted  is 
proportional  to  the  temperature. 

We  can  also  prove,  as  we  will  hereafter,  that  this  principle 
holds  good  when  the  curves  T^T^,  T,T^,  T-.T-^  are  not  iso- 
thermals,  but  simply  curves  of  the  same  kind,  that  is,  which 
follow  the  same  law  of  change  of  pressure  and  volume,  lohat- 
ever  that  law  may  he. 

From  the  proportion 

Q::  Qr.  Qs=T,:  T,'.T„ 
we  obtain 

-^=-^  =  ^,etc (XXV.) 

Equations  for  the  Expansion  of  Air  under  constant  Pressure.— 
Let  us  now  consider  the  case  in  which  the  specific  air-volume 
expands  under  constant  pressure,  while  it  is  heated.  The  heat 
imparted  serves  here,'  1,  to  increase  the  vihration  worJc,  and 
2,  to  perform  outer  ivorJc. 

In  the  beginning,  let  us  have  the  volume  Vi  at  the  tempera- 


176  THERMODYNAMICS. 

ture  ^1  and  pressure  ^h-     Af ier  expansion,  let  the  volume  be  v, 
and  the  temperature  t.     Then  the  increase  of  vibration  work  is 

Let  now  0?;,  =  -Wi  =  the  initial  volume,  Fig.  17,  and  Ov  — 
V  =  the  final  volume,  then  the  outer  work  performed  (ZJ  will 
be  represented  by  the  rectangle  v^ABv,  or  by  the  difference  of 
the  products  * 

pv  —  piVi  =  piP  —  piVi , 
since  the  pressure  is  constant,  and  hence  p  =  p^. 

We  have,  therefore, 

L,=p{v-ih)-=p,{v-v,)   .    .      (XXVI.) 
or  since 

L,  =  B{t~t,).f.     .     .     .     (XXVII.) 

By  means  of  these  two  equations,  we  can  calculate  the  in- 
crease of  the  inner  work  and  the  outer  work  performed. 

As  in  general  the  ratio  be- 
Y  tween  the  initial  and  final  tem- 

peratures is  given,  we  can  easi- 
ly determine  the  temperature  t 
A  B  ^^  ^-^^  ^^^  ^^  expansion.    Thus, 

/!~~~— ^-r-"-T-T^\  as  we  have  already  shown, 

/:   ■  \  273  +  t      273 +^i' 


M 


or 


«i£i_273_+^ 
vp  ~  273  +  r 
and  since,  in  the  present  case,  p  =  p^,  we  have 
_v_  _  273  +  t 
V,  ~  273  +  ti  ' 

EXAMPLE. 
Suppose  in  a  cylinder  one  unit  of  weight  of  air  at  a  temperature  ^=30°. 
Wliat  temperature  will  it  possess  when,  being  heated,  it  expands  under  constant 

*  Any  curve  or  line,  like  AB,  which  represents  the  states  of  a  body  when  the  pressure 
remains  constant,  is  sometimes  called  an  "  isopiestic  line  "  or  curve,  that  is,  a  line  or  curve  of  equal 
pressure.  Such  a  line  is  also  sometimes  called  an  "isobar.'''  In  lilce  manner,  a  line  which 
represents  the  states  of  a  body,  when  the  volume  is  constant,  is  sometimes  called  an  "  isometric 
line,"  or  line  of  equal  volume, 

t  If  there  are  Q  kilograms,  we  have  L  =  GJi{t  -  ti). 


EXPANSION  OF  AIR.  I77 

pressure  until  it  is  ^ds  of  its  original  volume?    What  outer  work  wiU  it  perform? 

and  what  is  the  increase  of  its  inner  work? 

Since  v  —  ^Vi,  we  have 

iv,_m_+±  ^^303-273  4-/ 

1^-373T30'         .x303-373  +  ^, 

if  =  404  -  273,     or    t  -  131°. 
The  increase  of  inner  work  is 

Z7=  ^  (131  -  30)  =  71.431  x  101  =  7214.53  meter-kil., 
and  the  outer  work  performed  is 

L  =  B  (131  -  30)  =  29.272  x  101  =  2956.47  meter-kilograms. 
The  amount  of  heat  consumed  in  each  work  may  be  easily  found.  We  know 
that  the  specific  volume,  when  heated  under  constant  pressure,  requires  0.23751 
heat  units,  or  1.41  times  as  much  as  when  the  air  is  heated  under  constant  vol- 
ume. Since  we  denote  the  specific  heat  for  constant  volume  by  c,  that  for  con- 
stant pressure  is  1.41c  for  air,  or,  in  general,  kc. 

If,  therefore,  we  heat  one  kilogram  of  air  t  —  t^  degrees  under  constant  press- 
ure, we  have 

Q  =  JcG{t-  t,) 

heat  units  necessary  to  be  imparted. 
In  the  present  case,  then, 

Q  =  0.23751  (131  -  30)  =  23.9885  heat  units. 

The  preceding  enables  us  to  find  an  expression  for  the  mechan- 
ical equivalent  of  heat  in  a  more  general  method  than  before. 

The  quantity  of  heat  Q,  in  the  last  expression,  is  equivalent 
to  the  work  AL,  hence 

AL  =hc{t-  t,)     or    L  =  ^{t-  t,). 

But  we  must  have  this  work  L  equal  to  the  vibration  work 
plus  the  outer  work,  or 

L=  U  +  L^,     or 

If  we  divide  through  hj  t  —  t^,  we  have 

1   _        B 

A        c{k-l) 
Inserting  numerical  values  as  determined  for  air,*  we  have 
1  29.272  29.272      ,^^  ^       ,     ,  ., 

-A  =  0.16847  X  0.41  "  0:0607  =  ^^^'^  meter-kilograms. 

*  Page  146. 

12 


178  THEBMODTNAMICS. 


QUESTIONS  FOU  EXAMINATION. 

What,  do  j'ou  understand  by  an  isothermal  curve  or  lino  ?  What  is  the  law  of  this  curve  ? 
What  kind  of  a  curve  then  is  it  ?  What  do  you  mean  by  "  permanent "  gas  ?  Are  there  any 
such  ?  What  is  the  graphical  representation  of  Mariotte's  law  ?  Find  the  expression  for  the 
mechanical  work  of  gas  expanding  according  to  this  law.  Find  the  expression  for  the  amount 
of  heat  imparted. 

What  do  you  understand  by  an  Isodynamic  curve  or  line  ?  When  is  this  curve  the  same  as 
the  isothermal,  and  for  what  kind  of  bodies  ?  What  is  an  isenergic  curve  ?  What  does  "  isen- 
ergic  "  mean  ?    What  do  you  understand  by  energy  ? 

What  is  an  adiabatic  curve  ?  Isentropic  curve  ?  What  does  "  isentropic  "  mean  ?  Define 
"entropy."  What  does  " adiabatic "  mean  ?  What  is  the  law  of  the  adiabatic  curve  ?  Deduce 
it.  Why  has  it  been  called  the  exponential  law  of  Mariotte  ?  In  what  respect  is  the  term 
appropriate?  Which  curve,  the  adiabatic  or  the  isothermal,  approaches  the  axis  of  X  most 
rapidly  ?  Wtiy  ?  Deduce  from  the  general  law  a  relation  between  volume  and  temperature. 
Between  pressure  and  temperature. 

Show  how  to  determine  the  outer  work  performed  by  air  expanding  adiabatically.  To  what 
is  the  outer  work  solely  proportional  in  this  case  ?  Why  might  this  have  been  at  once  con- 
cluded ?  Deduce  the  expression  for  the  heat  units  converted  into  work.  Discuss  in  similar 
manner  the  case  of  air  when  compressed  adiabatically. 

When  a  gas,  as  air,  is  made  to  pass  according  to  any  given  law,  from  one  adiabatic  curve  to 
another,  what  is  the  relation  between  the  heat  imparted  and  the  temperature  ?  Prove  this 
relation. 

What  is  an  isopiestic  line  ?  What  does  "  isopiestic  "  mean  ?  When  air  expands  under  con- 
stant pressure  what  effects  does  the  heat  imparted  produce  ?  What  is  the  expression  for  the 
increase  of  vibration  work  ?  What  is  the  expression  for  the  outer  work  performed  ?  What  is 
the  relation  between  volume  and  temperature  ? 

If  one  kilogram  of  air  has  a  temperature  of  30°  C,  what  will  be  its  temperature  when  it  is 
heated  and  made  to  expand  under  constant  pressure  until  it  is  twice  its  original  volume  ?   What 
outer  work  will  it  perform  ?    What  is  its  increase  of  inner  work  ?    What  amount  of  heat  is  im- 
parted ?     Prove  that  —  =  — ^ —  . 
^  A       c(k-\) 

What  work  is  performed  by  10  pounds  of  air  at  2  cubic  feet  volume  and  5  atmospheres  press- 
ure, when  it  expands  to  7  cubic  feet,  overcoming  an  outer  pressure  equal  at  an^'  moment  to  the 
tension,  the  temperature  being  kept  constant  ? 

What  IS  this  constant  temperature  'i  What  is  the  final  pressure  ?  How  much  heat  must 
have  been  imparted  in  order  to  keep  the  temperature  constant  ? 

If  one  kilogram  of  air  is  heated,  under  the  pressure  of  the  atmosphere,  from  0°  to  1°  C,  how 
much  work  does  it  perform  during  expansion  ? 

If  one  pound  of  air  is  heated,  in  same  manner,  from  32°  to  33°  Fahr.,  what  work  is  performed? 
If  heated  from  0°  to  1°  C,  what  work  is  performed  ? 

What  is  the  weight  of  one  cubic  foot  of  air  at  atmospheric  pressure  and  32°  F.  ?  What  is 
weight  of  one  cubic  meter  of  air  at  same  pressure  and  0°  C.  ? 

If  two  pounds  of  air  at  a  temperature  of  40°  Fahr.  expand  adiabatically,  performing  work, 
till  the  volume  is  doubled,  what  is  the  final  temperature? 

What  is  the  original  volume  ?  What  is  final  pressure,  if  the  initial  pressure  is  one  atmos- 
phere ?    What  is  the  work  performed  ? 


CHAPTER  VI. 

THE  SIMPLE  EE7ERSIBLE    CYCLE    PROCESS." — ILLUSTRATION    OP   THE 
PROCESS   BY  ANALOGOUS   PRINCIPLES   OF   MECHANICS. 


Suppose  in  the  cylinder  CC,  Fig.  18,  one  nnit  in  weight  of  air 
of  tlie  volume  Vi,  tension  j)\  and  absolute  temperature  T^. 

Let  this  volume  ex- 
pand, performing  work 
(the  outer  pressure  be- 
ing always  equal  to  the 
tension,  or  differing  only 
by  an  infinitely  small 
amount),  until  it  has  the 
volume  Ov^=  Vc,,  and  the 
tension  p^.  Let  us  also 
assume  that  the  temper- 
ature Ti  remains  the 
same,  so  that  heat  must  be 
imparted  from  without. 
The  expansion  therefore 
takes  place  along  the  iso- 
thermal line  T^Ti,  and 
the  heat  imparted  is  Qi. 
The  area  ViTiTiVc,  rep- 
resents the  work  per- 
formed by  the  air  during 
expansion. 

Now  let  the  air  still  expand  from  the  volume  Ov,  =  v.,  to 
Ovj  =  v.,  still  performing  work,  but  without  receiving  or  part- 

*  [A  c)-cle  is  termed  ''closed  "  if  the  body  after  any  series  of  changes  returns  to  its  original 
condition,  otherwise  it  is  called  "o;je«."  A  closed  cycle  is  termed  ''simple"  if  its  bounding 
curves  are  of  only  two  types.  If  of  more  than  two  types,  it  is  called  compound.  A  closed  cycle 
process  is  "  reversible  "  when  the  changes  of  state  are  continuous.  In  such  a  cycle  all  the  changes 
may  take  place  in  reverse  order.] 

179 


180  THERMODYNAMICS. 

ing  with,  lieat,  that  is,  let  it  ex23and  adiabatically.  The  ex- 
pansion then  follows  the  acliabatic  curve  T^T^,  and  the  area 
TiT^v^v-i  represents  the  outer  work  performed.  Since  this 
work  must  be  at  the  expense  of  the  vibration  work  or  tempera- 
ture, this  latter  must  decrease,  and  T^  falls  to  T2. 

The  work  thus  far  performed  is  therefore  given  by  the  area 
T^T^T^v^v^Tx,  and  the  amount  of  heat  imparted  is  Qx  heat 
units. 

Let  now  the  air,  whose  volume  is  OV3  =  v^  and  temperature 
T'a,  be  compressed,  the  temperature  T^  being  kept  the  same,  until 
its  volume  is  0^4  =  v^.  During  this  process  a  certain  amount 
of  heat  Q.  must  be  abstracted  in  order  that  the  temperature 
may  remain  the  same.  The  compression  follows  then  the 
isothermal  line  T^T,,  and  the  mechanical  work  necessary  for 
compression  is  represented  by  the  area  T^v.ViTi. 

The  volume  OVi  —  v^  may  be  so  chosen  that  when  the  air  is 
finally  compressed  adiabatically  to  the  volume  Vi,  the  air  shall 
be  again  in  its  initial  condition.  In  this  case,  the  work  per- 
formed upon  the  air  is  given  by  the  area  ViT^T-^v^,  the  tem- 
perature rises  from  T^  to  T^,  and  the  expansive  force  rises  from 
Pi  to  px  again. 

A  process  of  this  kind  is  called  by  Clausius  a  "  simple  reversi- 
ble cycle  process^  * 

As  remarked,  the  work  done  by  the  air  is  given  by  the  area 
TiT^T^v^v^T-x,  and  that  performed  upon  the  air,  or  used  in 
the  compression,  is  given  by  the  area  T'lV^ViTxT'^T^.  The 
difference  of  these  two  areas,  or  the  area  T^T^T^T^,  which  is 
shaded  in  the  Fig.,  and  is  inclosed  by  the  two  isothermal  and 
adiabatic  curves,  is  the  excess  of  the  loorh  performed  by  the  air  over 
that  performed  upon  it. 

Since,  however,  heat  and  mechanical  work  are  equivalent ; 
since  the  generation  of  work  requires  heat,  and  the  generation 
of  heat  requires  work  ;  the  heat  Q^  imparted  during  the  change 
from  Ti  to  J'l,  as  shown  by  the  arrow,  must  be  greater  than 
the  heat  abstracted,  Q^,  which  is  also  indicated  by  an  arrow. 
In  other  words,  the  difference  of  the  heat,  units  Qi  —  Q^,  trans- 
formed into  work,  is  represented  by  the  area  T^  Tx  T^  T^. 
Since  the  heat  Qx  —  Q^  is  equivalent  to  the  work 

*  See  precedinn;  note. 


INTERMEDIATE  BODY  IN  THE   CYCLE  PROCESS.       181 
we  have,  if  we  denote  tlie  area  Tj  T^  T^  T^  by  F, 

^{Q,-Q2)=F  ....     (XXYIII.) 

or 

Q,-Qz=--  AF. 

"We  have  thus  far  assumed  that  the  initial  condition  of  the 
air  is  given  by  the  quantities  v^ihT-i,  and  that  this  condition 
passes  gradually  into  the  condition  T^,  p^,  v^,  or  Tj,  p^,  v^,  or^ 
what  is  the  same  thing,  that  the  end  of  the  ordinate  ^i  traverses 
the  outline  of  the  area  T.T.F.T^in  the  direction  1\T^T,T,T^. 
We  may  have  its  motion,  however,  in  the  opposite  direction, 

In  such  case  we  have  evidently  the  work  of  compression 
greater  than  that  performed  by  the  air  during  the  expansion . 
and  the  difference  of  the  two  is  still  given  by  the  area 
T,T,T^T,. 

Since,  then,  work  is  performed  upon  the  air,  an  equivalent 
amount  of  heat  is  generated  in  it,  or,  in  other  words,  the  heat 
Qi  abstracted  during  the  change  T^T^  must  be  greater  than  the 
heat  Q^  imparted  during  the  change  T.T..  We  have  therefore 
obtained  heat  instead  of  work. 

Since,  then,  mechanical  work  has  disappeared,  and  heat  been 
obtained,  we  must  have  the  same  equation  as  before,  but  with 
opposite  sign.     Thus 

-^{Q.-Q'^^-F .    .    .    (XXIX.) 

or 

-{Qx-Q-^^-AF. 

In  fact,  the  arrows  in  the  Fig.  must  now  have  an  opposite 
direction;  the  lower  should  point  toward  the  curve  T.2T:,  the 
other  away  from  T^T^. 

The  reason  for  the  change  of  sign  is  also  seen  in  that,  in  the 
present  case,  where  heat  is  generated,  the  end  of  the  ordinate 
pi  goes  around  the  area  i^in  a  direction  opposite  to  that  in  which 
it  went  w^hen  heat  disappeared  and  work  was  given  out.  Since 
the  above  process  can  be  performed  in  either  way,  we  call  it  a 
"■simple  EEVEESIBLE  cycle  process.'''' 

Intermediate  Body  in  the  Cycle  Process. — It  is  evident  that  for 


182  THERMOBTNAMICS. 

tlie  performance  of  a  cycle,  we  must  have,  in  addition  to  the 
body  wliicli  goes  through  the  cycle  of  changes,  and  which  we 
may  call  the  "working  body"  or  the  '' intermediate  body,'''  also 
two  other  bodies,  one  of  which  gives  out  and  the  other  of  which 
absorbs  heat.  In  the  hot-air  engine  the  fire  is  the  source  of 
heat,  the  air  in  the  engine  is  the  intermediate  body,  and  the  out- 
side air  is  the  body  which  absorbs  heat,  or  the  refrigerator.  In 
the  steam  engine  the  water  is  the  intermediate  body,  the  hot 
gases  of  combustion  the  source,  and  the  cold  condensing  water 
or  the  outside  air  the  refrigerator. 

If  the  absorption  and  rejection  of  heat  by  the  intermediate 
body  takes  place  as  represented  by  the  cycle  diagram,  it  is  evi- 
dent that  the  source  of  heat,  which  we  denote  by  K,  must  pos- 
sess so  much  heat  as  to  replace  each  time  the  loss  of  heat  of 
the  intermediate  body,  and  that  this  latter  body  must  be  able 
to  at  once  receive  each  new  accession. 

In  like  manner  the  refrigerator,  which  we  denote  by  K^,  must 
be  able  at  every  moment  to  absorb  the  heat  excess  from  the  in- 
termediate body. 

In  practice  these  conditions  are  seldom  perfectly  fulfilled, 
and  hence  we  generally  find  the  calculated  work  varies  more  or 
less  from  that  obtained,  by  direct  observation. 

Now  that  we  have  introduced  the  bodies  K  and  K^,  let  us  con- 
sider once  more  our  cycle  process. 

While  the  end  of  the  ordinate  pi  describes  the  circumference 
of  the  shaded  area  in  the  direction  J\  T^  T,,  etc.  The  inter- 
mediate body,  or  in  this  case  the  air,  receives  the  heat  Qi 
while  passing  from  T^  to  T^.  The  source  K  then,  which  imparts 
this  heat  ^i,  sinks  in  temperature.  While  the  intermediate  body 
passes  from  T.  to  T-^,  it  gives  up  the  quantity  of  heat  Q.,  to  the 
refrigerator  iTi,  and  hence  the  temperature  of  K^  rises.  But  now 
the  heat  Q^  is  greater  than  Qi  and  the  excess  corresponds  to  the 
work  performed  by  the  intermediate  body.  The  body  K,  there- 
fore, imparts  more  heat  than  the  body  K^  receives.  The  disap- 
pearing heat  is  transformed  into  work. 

In  a  steam  engine  then,  where  the  steam  is  the  intermediate 
body,  the  heat  luhich  the  steam  possesses  before  joerforming  the  work 
must  be  greater  than  that  tvhich  it  possesses  after,  and  which  it  im- 
parts to  the  condensing  ivater  or  to  the  air,  and  the  difference,  trans- 
formed into  work,  acts  tipon  the  pdston.     The  same  holds  good  for 


THE  CYCLE  PROCESS.  183 

tlie  hot-air  engine.  Tlie  Jieat  wliicli  tlae  hot  air  gives  up  to  the 
outer  air  is  less  than  the  heat  imparted  to  it,  by  an  amount 
equivalent  to  the  work  performed. 

Carnot,  to  whom  the  cycle  process  is  due,  laid  down  the  fol- 
lowing principle  :  "  When  heat  passes  from  one  body  K,  through 
an  intermediate  body,  to  a  third  K^,  work  is  performed,  but  the 
heat  received  by  K^  is  equal  to  that  lost  by  K.''  This  last  clause 
contradicts,  as  we  see,  the  fundamental  principle  of  thermo- 
dynamics, and  is  incorrect. 

If  the  end  of  the  ordinate  jh  passes  round  the  shaded  area  in 
the  direction  T^  T^  T„,  etc.,  then  along  the  path  Tj  T^,  the  heat 
Qi  is  absorbed,  and  the  temperature  of  K^  sinks.  Along  the 
path  Ti  T^  the  heat  Q^  is  rejected,  and  this  heat  Qi  is  greater 
than  Q,  by  the  amount  of  heat  equivalent  to  the  work  performed 
upon  the  intermediate  body.  This  heat  ^i  —  Q^  is  now  given 
up  to  the  body  K,  which  thus  receives  more  heat  than  the  first 
gave  out. 

Transformation  of  Equation  XXVIIL — We  can  now  give  to 
the  Equation, 


A 
or 

Q^~Q,  =  AF (1). 

another  form. 

Since  the  curves  T^  Ti  and  T^  T^  are  adiabatic  curves,  and 
Tx  Tx  and  Tj  T.^  are  isothermal  curves,  we  may,  according  to 
Eq.  XXV.,  express  the  ratio  of  the  quantities  of  heat  Qi  and 
^2,  in  terms  of  the  absolute  temperatures. 

We  have,  according  to  that  Equation, 


therefore 


T,       T,' 


If  we  substitute  the  value  of  Qi  in  (1)  we  have 

q,-^^  =  AF,     or     Mi^_M:^  =  Ai^, 


184  THERMODYNAMICS. 

or 


If  we  substitute  the  value  of  Q^  in  (1)  we  have 

9^1 -Q^  =  AF,   or   ^'^'  -  ^'^'  =  AF, 


F  =  -^^{T,-T,).     .     .     .     (XXXI.) 
Since  -^  =  424  meter-kilograms,  we  have  also 

We  have,  therefore,  the  following  important  principle :  Tlie 
work  performed  by  the  intermediate  body  in  a  simple  reversible 
cycle  process,  or,  when  the  process  is  reversed,  the  worh  performed 
upon  this  body  is  directly  proportional  to  the  amount  of  heat  absorbed 
or  rejected,  and  also  to  the  difference  between  the  higJiest  and  loivest 
temperatures. 

The  correctness  of  this  principle  is  also  seen  at  once  from  our  , 
Figure.  The  greater  J'l  or  the  less  T^,  or  the  higher  the  end  of 
the  ordinate  pi  and  the  lower  that  of  p-^,  so  much  the  greater  is 
the  vertical  depth  of  the  shaded  area.  Also,  the  more  the  gas 
expands  under  constant  temperature  along  the  isothermal  T^^T^ 
— the  more  heat,  therefore,  is  imparted  to  it  in  order  to  main- 
tain its  temperature  constant — the  further  apart  are  the  ordi- 
nates  p^  and  p^,  and  so  much  the  greater  is  the  length  of  the 
shaded  area. 

The  heat  Q^  absorbed,  or  the  heat  Qi  rejected  by  the  gas  in 
passing  over  the  isothermals  T^  T^  or  T^  T 2,  may  be  calculated 
from  Equation  XY.,  when  the  initial  and  end  volumes,  Vj  and  v, 
and  the  initial  and  end  pressures,  p)^  and  p),  are  known.  But 
from  Equation  XVI.  we  can  determine  Q^  and  Q^,  when  we 
know  the  initial  and  end  volumes,  as  well  as  the  temperatures 
Ti  and  T^. 


TEE  SIMPLE  CYCLE  PROCESS.  185 


EXAMPLE. 

A  hot-air  engine  is  so  constructed,  tliat  each  unit  of  air  in  it  makes,  for  each 
double  stroke  of  the  engine,  a  complete,  simple,  reversible  cycle  process.  The 
initial  temperature  T^  is,  for  the  greatest  compression,  273  +  300  =  573%  the 
lowest  temperature  T.,  is  273°  +  0  =  273\  What  work  will  the  air  perform 
Avhen  it  expands  from  the  initial  volume  o^  to  i|  v-^  under .  constant  temperature, 
and  then  expands  adiabatically  ? 

The  heat  imparted  is,  from  Equation  XVI., 


Q,  =  2.3026  ^i^^i  lo; 
29.273  X  573  x  0.125 


=  11.39  heat  units. 


424 
The  mechanical  work  is  therefore,  from  Equation  XXX., 

O  to 

=  8.43  X  300  =  2529  meter-kilograms. 
If  we  substitute  in  tlie  formula 

tiie  value  of  Q^  from  XYI.,  we  have 

F  =  2.3026B  (T,  -  T2)  log.  -  .    .    (XXXII.) 

The  mechanical  w^ork,  therefore,  increases  with  the  expansion 

ratio    —    and  the  difference  of  the  absolute  temperatures  T^ 

and  T2. 

Since  T^  -  T^  =  (273  +  t^-  [273  +  ^2]),  we  have  also 

F  =  2.3026^  (fi  -  4)  log  - .     .     (XXXIII.) 

where  t-^  and  1^2  are  the  temperatures  in  centigrade  degrees-  We 
shall  have  occasion  to  refer  to  these  equations  in  our  discus- 
sion of  the  engines  of  Ericsson  and  Lehmann. 

Illustration  of  these  Principles  hy  Analogous  Mechanical  Princi- 
ples.— In  illustration  of  the  formula 


186 


THEBMOD  TNAMIG8. 


F. 


Q2 


Al 


t{T,-~T,) 


we  may,  as  lias  been  done  bj  Zeuner,  make  use  of  tiie  follow- 
ing meclianical  considerations. 

Let  AAi,  BBi,  and  CC\  be  three  planes  one  above  another. 
The  distance  between  AA^  and  CC^  is  h,  and  between  BB^  and 
AA^,  Ih. 

Suppose  a  weight  G  placed  upon  BB^,  then  the  uniform 
sinking  of  this  weight  through  the  height  7^2  will  perform  the 
work 

G\. 

If,  for  example,  G  is  the  weight  of  a  quantity  of  water  which 
arrives  every  second,  the  fall  of 
this  water  may  put  in  motion  a 
vertical  water-wheel,  and,  for 
uniform  motion  of  the  wheel, 
give  a  mechanical  effect  every 
second  expressed  by 


\c 


B 

r 


-A, 


Glh. 

If  we  denote  this  product  by 
^2j  we  have 


F^  =  Glh,  or 


G  = 


(1). 


If  the  same  mass  were  to  sink,  performing  work,  from  the 
plane  COi,  we  should  obtain  the  work  Gh.  If  we  denote  this 
by  i^i,  we  have 

F, 


F^  =  Glh,    or    G  = 


K' 


(2). 


If  we  raise  the  weight  G  from  the  plane  BB^  to  the  plane 
CCi,  the  work  performed  is 

F--=Gih-h). 

This  would  also  be  the  work  obtained  if  we  should  allow  G 
to  sink  uniformly  from  the  upper  plane  to  the  lower,  and  then 
raise  it  with  the  same  uniform  velocity  from  the  lowest  plane 
to  the  plane  BB^. 


ANALOGOUS  MECHANICAL  PRINCIPLES.  187 

If  now,  we  substitute  in  tlie  last  equation  tlie  value  of  G  from 
(2)  and  (3j,  we  liave 

or 

These  equations  liave  j)recisely  tlie  same  form  as  those  al- 
ready given,  viz. : 

and 

There,  the  products  424  (^i  and  424^2  represent  the  mechani- 
cal works  corresponding  to  the  quantities  of  heat  Qi  and  ^2, 
just  as  now,  F^  and  F<i  represent  the  mechanical  works  obtained 
by  the  sinking  of  the  weight  G  from  the  heights  Ih  and  h^.  If 
we  allow  the  weight  G  to  fall  freely  through  the  distances  7^l 
and  h^,  the  products  Glh  =  F^  and  Glh  —  F^  will  represent  the 
work  potential  in  the  weight  in  the  form  of  living  force.     But 

if  the  products  424^i  and  424^2?  or  what  is  the  same  thing,  ^ 

and  ~  represent  mechanical  work,  we  may  consider  the  quo- 

tients 

424^,      424^2     _     Qt  .      Q2 


or 


and 


Ti    '       T,   '  AT,  AT2 

as  weights,  the  so-called  "  heat  iveights,"  of  Prof.  Zeuner.*  Since 
further,  the  difference  T^^  —  T^  is  equivalent  in  significance  with 
111  —  K,  we  may  call  the  difference   T,  —  T-^,  the  "  temperature 

fair 

*  [The  '■'■heat weight^'  of  Prof .  Zeuner  is  identical  with  iho.^' thermodynamic funcUon'''' of 
Eanliine,  or  '■'entropy"  as  defined  by  Clausius.  The  term  '-entropy"  has  been  nsed  by  Tait, 
Thomson,  Maxwell,  and  others,  witli  an  entirely  different  signification.  The  term  "thermo- 
dynamic function  "  is  perhaps  good  enough  as  a  name  for  a  certain  function  of  the  heat  and 
temperature,  which  occurs  so  often  as  lo  render  a  special  name  for  it  desirable.  The  term  "  heat 
weight,"  not  only  answers  this  end,  but  also  gives  an  analogical  significance  to  the  term,  which 
s  of  real  service  in  using  it.  We  therefore  use  it  exclusively.  Those  who  prefer  to  call  it  "  ther- 
mic weight "  can  do  so.] 


188  THERMODTNAMIGS. 

We  may  therefore  express  tlie  principle  of  the  simple  revers- 
ible cycle  process,  as  follows  :  The  tvork  performed  by  the  inter- 
mediate body  in  the  simple  reversible  cycle  process,  is  directly  propor- 
tional to  the  heat  iceights  i  -~~  and  -~^  j  imparted  or  given  out, 
as  toell  as  to  the  temperature  fall  {T^—  T^^. 


QUESTIONS   FOR   EXAMINATION. 

What  is  a  cycle  proces^s  ?  When  is  sticli  a  process  said  to  be  "  closed  ? "  What  Is  a  simple 
cycle  process  ?  What  is  a  compound  ?  Wlien  is  a  cjxlc  process  reversible  ?  What  is  a  simple 
reversible  cycle  process  ?  By  how  much  d(ies  the  heat  imparted  exceed  that  abstracted  ?  Draw 
the  Figure,  and  show  what  represents  the  woik  performed.  What  is  the  intermediate  body  ? 
Give  examples.  In  a  steam  engine  is  the  heat  of  the  steam  before  performing  work  greater  tlian 
that  after  ?  What  l)ecomes  of  the  difference  ?  State  Carnot's  prmciple.  What  is  incorrect  in 
this  statement,  and  why  ?  Deduce  an  expression  for  the  work  in  a  simple  reversible  cj'cle  pro- 
cess in  terms  of  tlie  heat  absorbed  or  rejected,  and  the  highest  and  lowest  temperatures. 

Illustrate  these  principles  by  analogous  principles  of  meclianics.  What  is  the  heat  weight  ? 
Why  is  it  so  called  ?  What  is  the  temperature  fall  ?  How  do  you  express  the  work  in  the  simple 
reversible  cjxle  process  in  terms  of  the  heat  weight  and  temperature  fall  ?  What  do  you  under- 
stand by  thermodynamic  function  ?  What  other  term  has  beeu  given  to  this  quantity  ?  Why  is 
it  not  appropriate  ? 


CHAPTEE  VII. 

GENERAL  LAW  OF  THE  RELATION  BETWEEN  PRESSURE  AND  VOLUME 
OF  A  GAS — GRAPHICAL  REPRESENTATION  OE  THE  INNER  WORK. 

Revieiv  of  the  preceding  Principles. — We  have  thus  far  con- 
sidered the  following  cases  of  the     v 
change  of  pressure  and  volume  of 
a  gas,  especially  of  air. 

1.  The  volume  is  constant,  and 
hence  the  expansive  force  in- 
creases with  the  temperature. 

If  Ov  is  the  specific  volume  at 
0°  and  p  the  pressure,  then  the 
pressure  at  273°  is  2p,  etc.  The 
line  vA,  which  gives  the  relation  ^i«-  ^o- 

between  pressure  and  volume,  is  parallel  to  the  axis  of  ordi- 
nates  OZ* 

If  we  denote  the  inner  work  at  the  temperature  T^  by  U^,  we 
have 

U,  =  ~T,c, 

where  c  is  the  specific  heat  for  constant  volume. 

If  U  is  the  inner  work  for  the  temperature  T,  we  have 

U=\  Tc, 

A 

hence  the  change  of  inner  work  is 


0 


X 


U'  u^ 


(T-T,). 


*  [Such  a  line  may  be  called  an  ' 


sometric  Ime,"  or  line  of  equal  volume.] 

189 


190  THEBM0DYNAMIC8. 

Or,  since  r=  273  +  ^  and  T,  -  273  +  k, 

If  Q  is  the  lieat  imparted  during  this  change, 

or 

Q  =  c{t-t,). 

According  to  the  combined  law  of  Mariotte  and  Gay-Lussac, 
we  have 

pv  =  BT,     and    2hVi=-BT^; 
hence 

or,  since  Vi  =  v, 

and  hence 

T        273  +  ^ 
^'=^^I;=^273T"^/ 

2.   The  expansive  force  is  constant,  or  the  expansion  takes  place 
under  constant  pressure. 

If  Ovi  Fig.  21  is  the  specific 
Y  volume  for  the  temjDerature  t^ 

and  pressure  pi,  and  Ov  the 
volume  for  the  same  pressure 
A,  '"B  p  and  the  temperature  t,  then 

the  change  of  inner  work  is 


0 


U=^^{t-t,l 


"-    and  the  line  AB  parallel  to  the 
axis  of  abscissas  gives  the  re- 


lation between  volume  and  pressure. 


*  [Such  a  line  may  be  called  an  "  isopiesiic  line"  or  line  of  equal  pressure-] 


REVIEW  OF  PRECEDING  PRINCIPLES. 


191 


Tlie  outer  work   performed,  wliicli  is  represented  by  tlie 
rectangle  v^ABv,  has  the  value 

Li=p{v  —  Vi), 
or  L,  =  B{t-  t,). 

Since  here  both  outer  and  inner  work  is  performed,  we  have 
the  equation 

L=  U+L„ 

or  L  =  ^{t-t,)  +  B{t-  t,)  =  (^  +  ^)  (^  -  ^i)- 

The  heat  imparted  is     • 

Q  =  Jcc{t-t,), 
where  kc  is  the  specific  heat  by  constant  pressure. 
Further,  we  have 

p,v^  =  BT, 

pv=BT,    hence,     |^  =  ||, 

or  since  pi  =  jo, 

V  _  T  _  273  +  ^ 
v,~  T,~273  +  h' 


3.  The  air  expands  under  constant  temperature,  according  to 
Mariotte's  law : 

pv  =  20iVi  =  PiV^  =  ^93^3,  etc. 

In  this  case  the  end  of  the  ordi- 
nate p,  which  gives  the  pressure 
for  the  volume  v  and  temperature 
t,  describes  the  isothermal,  or,  in 
the  case  of  a  gas,  the  isodynamic 
curve*  AB,  and  the  area  of  the 
space  ViABv,  gives  the  outer  work 
performed. 

This  we  have  found  to  be 


L  =  2.3026  v^p,  log 


vi- 


X  =  2.3026  Vi«i  log  ^. 


[Sometimes  called  the  '"isenergic  curve.'"'\ 


192  THERMODYNAMICS. 

The  quantity  of  lieat  imparted  is 

Q  =  2.3026  Av^p^  log  - 


Q  ^  2.3026  Ai)p  log  ^ 


We  have  also  found 


and 


Q  =  2.3026  ART,  log  ^  ,  etc., 


i;  =  2.3026^^  log  —  ,  etc. 


While,  therefore,  in  (1)  and  (2)  the  heat  Q  imparted  may  be 
found  directly  from  the  initial  and  final  temperatures,  or,  as 
we  say,  is  a  function  of  t,  and  t,  here  it  is  determined  by  three 
quantities,  viz.,  the  temperature  T  and  the  initial  and  end  vol- 
umes, or  by  T  and  the  initial  and  end  pressures. 

4.  The  air  expands,  performing  ivorh,  at  the  expense  of  its  own 
heat,  according  to  the  law 


'R 


The  end  of  the  ordinate  p,,  Fig. 
23,  which  gives  the  initial  pressure 
for  the  sjDecific  volume  v,  and  the 
temperature  ^i,  describes  during  ex- 
pansion the  abiabatic  curve*  AB, 
which  approaches  the  axis  of  ab- 
scissas OX  more  quickly  than  the 
isothermal. 

Further  we  have  found 


vV''^  273  +  ^1 
vj     ~273  +  t'' 


273  +  t, 
273  +  r 


*  [Sometimes  called  tlie  "  iseiitropic  cu7've,'"  or  curve  of  equal  entropy.] 


REVIEW  OF  PBECEDING  PBINCIPLE8.  193 

The  inner  work  which  disappears  is 

where  ^i  is  the  initial  and  t  the  final  temperature.     Since  this 
is  equal  to  the  outer  work  performed,  we  have  also 

and  this  work  (or  the  disappearing  inner  work),  is  represented 
by  the  area  i\ABi\ 
For  Q  we  have 


Finally,  we  have  also  found 


T-     ^ 


T  ^ 


c  1 

Since  we  have  found  -^-^  =  v — y ,  we  have  also 


^=.^i'-[-(^)'l 


General  Law  as  to  the  Relation  betiveen  Volume  and  Pressure. — In 
view  of  what  has  preceded,  it  will  now  be  easy  to  include  all 
these  special  cases  under  one  general  law. 

In  case  (4)  when  the  air  expanded  from  the  volume  Vi  to  v, 
there  v/as  no  heat  imparted,  the  pressure  diminished  rapidly 
as  the  volume  increased,  and  the  relation  between  pressure  and 
volume  was  given  by  the  equation 

PiVx'=P%'G2 (1). 

or,  in  the  case  of  air, 

In  the  following  Fig.  let  the  curve  ac  represent  the  law  of 
change  of  pressure  and  volume  for  this  case. 
In  case  (3)  as  the  volume  increased  the  decrease  of  pressure 
13 


194 


THERMOD  TNAMIC8. 


was  less  tlian  in  case  (4).     Tlie  law  of  change  was  given  by  the 
equation 

PiVx  =P'iT)2,  etc. 

This  equation  is  at  once  obtained  if  we  insert  ^  =  1  in  the 
first  equation  above.  Let  the  curve  ah,  the  isothermal  curve, 
represent  in  this  case  the  law  of  relation  between  the  pressure 
and  volume. 


Law  of (Jumffe  of  pressure'  wiih  Volume 


Compression 


ff(Mf 

hsorbecl 


In   case   (2)  the  pressure  was   constant  as  the  volume  in- 
creased, and  we  had 


P  =Pl  =P2, 


The  line  ad  or  ai 


and  this  obtained  from  (1)  by  making  Z;  = 
gives  the  law  for  this  case. 

In  the  first  case,  the  volume  remained  constant  while  the 
pressure  changed,  and  we  had 


l\  =  ?)2. 


This  is  obtained  from  Eq.  (1)  by  putting  0  for  the  exponent  of  • 
p  and  making  ^  =  1.     The  line  af  gives  the  law  for  this  case. 

We  see,  therefore,  that  the  laws  connecting  pressure  and 
volume  are  given  by  the  exponents  of  the  factors  v  and  p,  and 
for  different  exponents  we  may  thus  have  a  number  of  laws  and 


GENERAL  RELATION  BETWEEN  VOLUME  AND  PRE88UBE.  195 

curves,  all  of  wliicli  are  based  upon  tlie  same  general  law. 
If,  then,  —  is  any  number,  whole  or  fractional,  positive  or  nega- 
tive, we  have,  p^l^ ^9^ ^^i -  =  2?^ i\ k     .     .      (XXXIV.) 

or  p'"-  V'"  —  pr  ^\  =  Vi^  K 

is  the  general  laio  between  jjo^essicre  and  volume. 

Thus,  if  we  have  m  =  1  and  n  =  1,  we  have  the  law  of  the 
isothermal  curve  ab.  If  m  =  1  and  n  =  1.41,  we  have  the  adia- 
batic  curve  ac.  If  m  =  1  and  n  >  1.41,  we  have  a  curve  which 
approaches  the  axis  of  abscissas  more  rapidly  than  the  adia- 
batic.  Such  a  curve  is  al.  If  m  =  1  and  n  <1  but  greater 
than  0,  the  corresponding  curve  will  lie  between  ab  and  ad. 
If  01  has  a  negative  value,  if,  for  example,  m  =  1  and  n  -=  —  2, 
the  corresponding  curve  lies  between  ad  and  «/.  Such  a  curve 
is  represented  by  ae. 

All  these  curves  are  convex  to  the  axis  of  abscissas,  but  we 
may  have  curves  which  are  concave  also,  as  ao  or  ax. 

We  assume,  in  perfect  accordance  with  the  preceding,  that 
curves  to  the  right  of  the  line  Baf,  parallel  to  the  axis  of  ordi- 
nates,  apply  to  expansion.  In  such  case,  work  is  performed 
and  heat  is  absorbed,  as  indicated  on  the  Figure.  This  heat 
may  be  at  the  expense  of  the  heat  in  the  gas  itself,  or  it.  may  be 
imparted  from  without.  For  all  curves  between  ac  and  af,  on 
the  right,  heat  is  imparted  during  the  expansion.  On  the  other 
hand,  for  all  below  ac  heat  is  abstracted.  The  curves  left  of 
Baf  apply  to  compression.  In  such  cases  work  is  performed 
in  compressing  the  gas,  or  work  is  absorbed  by  the  inter- 
mediate body,  and  heat  is  generated.  In  the  Figure  ah  is  an 
isothermal  and  ag  an  adiabatic  line.  The  sign  +  indicates 
work  performed  by  the  body,  the  sign  minus  indicates  work 
absorbed,  or  performed  upon  the  body. 

The  above  considerations  find  a  very  interesting  application 
in  the  discussion  of  gas  engines. 

The  question  arises,  what  is  the  general  expression  which 
gives  the  general  law  for  the  temperature  when  the  relation 
between  pressure  and  volume  for  expansion  or  compression 
are  known  ?  and  how  can  we  determine  the  work  done  during 
expansion  by  the  intermediate  body,  or  absorbed  by  ifc  during 


196  THEBMODYNAMICS. 


We  can  put     ^i^^i'^  =  -  =i 2h^i^i'^ 


and  pv '"  = =  pvv 


lience 


that  is. 


or 


2hihVi-^     -- 

=pm 

7 — ■^ 

p{Gi=  RTy  and 

pv  = 

RT 

RT,tp-'  = 

--RTw^' 

a  +  t^  _  /v\^' 
a  +  t~  \vj 

-1 
=  1 

n 

-m 

il) 

= 

T 

(XXXV.) 


EXAMPLE. 

The  specific  volume  of  air  w,  =1  has  the  temperature  ty  =80°,  and  ex- 
pands performing  work  up  to  the  volume  v—2v-^.  What  will  be  the  final  tem- 
perature when  the  law  of  expansion  is^,i^i  —  ^  —pv—  ^  ? 

We  have 


273  +  30 

273TT-'    ^^^ 


(r"^=Q"^-^"' 


hence 

2^1^  =  ~     and    273  +  ^  =  303  X  8  r=  3424,    or 

t  r=  2424  -  278  =  2151°. 
If,  on  the  other  hand,  the  law  of  expansion  had  been 


we  should  have  had 

a 


/v'\  303  - 


a  +  t 

hence  2if  =  -  546  +  303  =  -  243,     or 

t  =  -  121.5°. 

The  curve  which  gives  for  this  case  the  relation  between  volume  and  pressure 
approaches  the  axis  of  abscissas  more  rapidly  than  the  adiabatic,  because  the 
exponent  2  is  greater  than  1.41,  and  as  already  explained,  we  must  therefore 
abstract  from  the  intermediate  body  a  certain  amount  of  heat.  (See  the  line  al 
in  the  Fig.) 


GENERAL  EXPRESSION  FOR  TEE  OUTER   WORE.        197 

If  in  tlie  formula  already  found  in  tlie  case  of  tlie  adiabatic 
curve. 


--.4Tm[i-e)'"*] 


we  put  h  =  — ,  we  have 
m 

and  this  is  the  general  expression  for  the  outer  work,  when 
the  relation  between  the  volume  and  tension  is  given  by  the 
general  formula. 

Since  now  (  -  )    '"■  =  ^  ,  andjpi'^i  =  HT^,  we  have  also 

n  —  m  L        TiJ      n  —  m      ^ 

n  —  m 
Also,  since 

c(^-l) 

L  =  ^?^  ^l^zil)  u  -t).    .   (XXXYII.) 
The  change  of  inner  work  is  as  always, 

If  Q  is  the  amount  of  heat  equivalent  to  these  two  works, 
we  have 

|-  =  Z7-  f/i  +  i, 

or 

Q  =  A{JJ--U,)+AL,    .     (XXXVIII.) 

and  by  inserting  the  values  above, 


198  THEBMODTNAMIGS. 


This 

may 

be  written 

Q:^C{t- 

«-« 

7)1 

^.{Jc- 

1)(^:- 

=  c{t- 

■  A)  [i  - 

m 

71  — 

-1)' 

_m'k  - 

■^c(t- 

-U) 

(XXXIX.) 

711  —  71^'  ^  ' 

This  formula  shows  that  the  heat,  Q,  imparted  or  abstracted, 
is  directly  proportional  to  the  difference  of  the  temperatures. 

But  the  amount  of  heat  Q  can  be  calculated  from  the  specific 
heat  and  the  temperature  difference  t^  —  t.  This  specific  heat 
is,  however,  unknown  in  the  present  case.  It  is  evident  that  it 
must  be  different  from  that  for  constant  pressure  or  for  con- 
stant volume.     Let  us  denote  it  by  s.     Then 

Q  =  s{t-t,) (XL.) 

If,  now,  we  put  these  two  values  of  Q  equal,  we  can  easily 
find  s.     Thus  we  have 

, ,       ,  -       mJc  —  71     ,,      ,  ^ 
sit  —  tA  = c(t  —  ti), 


5  = c (XLI.) 

If,  for  example,  the  law  of  change  of  pressure  with  volume 
is 

we  have  m  =  \  and  n  =  —  2,  and  hence  the  specific  heat  for 


this  law  of  expansion  is 


s  =  ^^  c,     or  for  air     — |i  x  0.16847  =  0.1915, 

that  is,  this  is  the  quantity  of  heat  which  must  be  imparted  to 
1  kilogram  of  air  when  expanding  according  to  the  assumed 
law,  in  order  to  raise  its  temperature  1°. 

The  curve  cdjc,  Fig.  25,  represents  the  law  in  the  present  case. 


GRAPHIC  BEPBESENTATION  OF  TEE  mNER    WORK.       199 


and  tlie  sliaded  area  acde  represents  tlie  outer  work  performed 
bj  tlie  gas  during  expansion,  when  the  specific  volume  increases 
from  Vi  to  |Vi.  "We  see  that  the 
curve  departs  rapidly  from  the  axis 
of  X  as  the  volume  increases.  It  is, 
evident,  also,  that  the  outer  press- 
ure gradually  increases  from  2^1  to 
20,  as  we  always  assume  the  outer 
pressure  at  any  moment  as  less  than 
the  inner  pressure  at  that  moment 
by  an  infinitely  small  amount. 


Gra2:)Mcal    Re2Dresentatiwi    of    tlie 
Inner    Woi^k. — We   can   now   repre- 
sent the  increase  or  diminution  of 
the  inner  work   in  a  manner  similar 
employed  for  the  outer. 


to  that  which  has  been 


Let  the  specific  volume  have  the  pressure  2h  and  the  vol- 
ume ^1,  and  let  the  inner  work-  be  C/j.  Then  let  the  air  ex- 
pand while  heat  is  imparted  to  it,  so  that  the  inner  work  Ui 
remains  constant.  The  expansion  takes  place  along  the  iso- 
thermal line  ade.    When,  then,  the  air  has  expanded  to  the 

condition  ^:>?i,  the  inner  work 
is  still  therefore  U-^. 

If  we  assume,  again,  heat  im- 
parted so  that  the  volume  i\ 
and  pressure  2^1  become  v^  and 
2^0,  the  change  taking  place  ac- 
cording to  any  law  or  curve,  as 
acb,  then  the  outer  work  per- 
formed is  Viabvz.  The  inner 
work  at  i\2h  is  then  no  longer 
f/"i.  Suppose  it  is  U2.  Let  us 
denote  the  heat  required  for 
this    change   by   ^1,  then  w^e 


have 


^1 


=  U2  -  Ui  +  Li, 


where  Li  is  the  outer  work  performed. 


200 


THERMOD  TNAMIC8. 


Let  us  now  assume  that  the  air  expands  at  the  expense  of  its 
heat,  until  its  volume  is  V  and  pressure  p.  The  expansion  fol- 
lows then  the  adiabatic  curve  hge,  and  the  area  hgeTiV2  gives  the 
outer  work,  or  what  is  the  same  thing,  the  disappearing  inner 
work.  Since  now  the  point  e  of  the  adiabatic  curve  falls  upon 
the  isothermal,  the  air  has  at  this  point  the  same  temperature 
as  at  any  point  of  ade.  The  inner  work  is  therefore  Ui.  While, 
then,  the  air  passes  from  the  condition  V2lh  to  the  condition  np, 
the  inner  work  changes  from  U^  to  U^,  or  the  inner  work  which 
disappears  is  Ui  —  U^.  Just  the  same  inner  work  has  been  im- 
parted on  the  way  from  'yj^:*!  to  T^Pz-  Hence  the  increase  of  the 
inner  ivork  ZZ^  —  Ux,  is  given  by  the  area  hgeWz. 

While  then  the  area  ViObv-i,  shaded  vertically  in  the  Figure, 
gites  the  outer  work  which  the  air  performs  in  passing  from 
Vipi  to  Vi'jh^  the  area  v-^bgev,  shaded  horizontally  in  the  Figure, 
gives  the  increase  of  inner  work. 

If,  therefore,  the  specific  air  volume  in  the  condition  v^p^ 
expands  under  the  addition  of  heat,  performing  work  along 
the  curve  acb,  until  the  volume  is  v^  and  the  pressure  pa?  we 
may  find  the  increase  of  inner  work  as  follows : 

Construct  through  a,  the  isothermal  ade,  and  the  adiabatic 
hge  through  h,  and  from  their  intersection  e,  let  fall  the  vertical 
ve.  The  area  vjbgev  gives  the  increase  of  inner  work.  The  area 
'dxabeT)  gives  the  sum  of  the  inner  and  outer  works,  which  is 

equivalent  to  the  heat 
Y  ~  imparted. 

The  increase  of  in- 
ner work  may  also  be 
found  as  follows  :  Pass 
through  h  the  isother- 
mal hgh,  and  through 
any  point  g  on  it,  the 
adiabatic  gi,  and  pro- 
duce it  till  it  meets  the 
isothermal  through  a 
in  i.  Let  fall  the  ver- 
tical lei.  Then  it  is 
at  once  evident  that  the  area  gikl  gives  the  inner  work  im- 
parted. 


GRAPHIC  BEPBESEI^TATION  OF  THE  IN^^ER   WORK.      201 


QUESTIONS   FOR  EXAMINATION. 

What  is  an  isometric  line  ?  Illustrate  for  air.  What  is  the  expression  for  the  change  of 
inner  work  ?  For  the  heat  imparted?  What  is  the  outer  worlt?  What  is  the  relation  between 
pressure  and  temperature  ? 

What  is  an  isopiestic  line  ?  Illustrate  for  air.  What  is  the  change  of  inner  work  ?  What  is 
the  outer  work  ?  What  is  the  heat  imparted  ?  What  is  the  relation  between  volume  aud  tem- 
perature ? 

Give  Mariotte's  law.  Illustrate.  What  is  an  isothermal  line  ?  An  isodynamic  ?  An  isen- 
ergic  ?  What  is  the  outer  work  for  air  ?  For  any  gas  ?  The  heat  imparted  ?  What  is  the 
change  of  inner  work  ? 

Give  the  adiabatic  law  for  air.  For  any  gas.  AVhat  is  an  isentropic  curve  ?  Why  may  we 
call  the  adiabatic  law  the  exponential  law  of  Mariotte  ?  What  is  the  inner  work  which  disap- 
pears ?  Why  ?  What  is  the  outer  work  ?  What  relation  does  this  bear  to  the  inner  work  ? 
Why  ?    Give  other  expressions  for  the  outer  work. 

State  the  general  law  between  ^•olnme  and  pressure.  Are  all  others  special  cases  of  this 
law  ?  What  changes  give  the  adiabatic  ?  The  isothermal  ?  The  isopiestic  ?  The  isometric  ? 
Illustrate  by  a  diagram.  W^hat  is  the  general  law  for  the  relation  between  volume  and  tem- 
perature for  any  perfect  gas  ?    Deduce.     Between  pressure  and  temperature  ?    Deduce. 

What  is  the  general  expression  for  the  outer  work  ?  Deduce  it.  Give  it  again  in  terms  of 
temperature.    What  is  the  general  expression  for  the  heat  imparted  ? 

Deduce  a  general  expression  for  the  specific  heat,  whatever  may  be  the  law  of  variation  of 
volume  with  pressure. 

Give  and  explain  graphical  representations  of  the  inner  work. 

If  10  cubic  meters  of  air  are  heated  under  atmospheric  pressure  from  0°  to  100°  C,  what  is 
the  new  volume  ?  What  is  the  new  density  ?  What  is  the  weight  of  each  cubic  meter  of  the 
new  volume  ?    What  is  the  work  of  expansion  ? 

If  heated  under  constant  volume,  what  is  the  new  pressure  ? 

If,  while  the  air  is  heated  and  expands,  the  temperature  is  kept  constant,  and  the  tension 
at  any  instant  is  equal  to  the  outer  pressure,  what  will  be  the  pressure  when  the  volume  is  12 
cubic  meters  ?    What  will  be  the  work  performed  ?    What  the  heat  imparted? 

If  no  heat  is  imparted,  what  will  be  the  pressure  when  the  volume  is  20  cubic  meters  ? 
What  will  be  the  work  performed  ?    What  amount  of  heat  will  disappear  ? 


NOTATION   OF  MOST  FREQUENT  USE, 


COMPILED   FOR 


CONVENIENCE  OF  REFERENCE. 


A  —  Z24  of  a  heat  unit  =  thermal  equivalent  of  one  unit  of  work. 

-T  —  424  meter-kilograms  =  mechanical  equivalent  of  one  unit  of  heat. 

a  —  co-efflcient  of  expansion  =  ^|-j  for  air  and  perfect  gases. 

c  =  specific  heat,  generally  for  constant  volume  unless  otherwise  specified. 

Cp  =  specific  heat  for  constant  volume. 

Cp  =  specific  heat  for  constant  pressure. 

D  —  density. 

F  —  outer  work  performed  by  or  absorbed  in  cycle  process. 

G  =  weight  of  a  given  volume  of  gas. 

J  =  disgregation  work  in  a  heated  body. 

k  =  -—■=  ratio  of  specific  heat  at  constant  pressure  to  that  at  constant  volume. 

Cv 

For  air  =  1.41. 
L  —  outer  work  performed  by  expanding  body. 
m,  n,  =  eo-effieients  of  pressure  and  volume  in  general  law,  for  variation  of  these 

quantities  in  gases,  viz.,  ^"^y"  =Pi  "^v-^"  . 
p  =  specific  pressure,  i.  e.,  pressure  upon  unit  of  surface. 
Q  =  amount  of  heat  measured  in  heat  units. 
H  —  outer  work  performed  by  expansion  of  one  unit  of  weight  of  gas,  when 

heated  under  pressure  of  atmosphere  (760'"''"'  =  10344  kil.  per  sq.  meter) 

from  0°  to  I'C. 

s  =  specific  heat  in  general,  whatever  be  the  law  of  variation  of  pressure  with 

mk  —  n 

volume  = c  . 

m  —  11 

T  =  absolute  temperature,  reckoned  from  —  273°  C. 

t  =  temperature  by  centigrade  thermometer. 

U  =  inner  work  =  vibration  work  plus  disgregation  work. 

V  =  volume  of  any  given  body. 

V  =  specific  volume,  i.  e.,  volume  of  unit  of  weight. 
W  —  vibration  work  when  any  body  is  heated. 

202 


RECAPITULATION   OF   PEINCIPAL   FOPtMUL^ 


CONTENIENCE  OF  REFERENCE. 


Fundamental  Equations  : 

Q  =  A{W+J+L) I.  (page  124.) 

Q  =  AiU+L) 11.  (page  124.) 

Expansion  of  gases — constant  pressure  : 

New  volume  =  V{1  +  0.003670, 

or     V(l  +  af) III.  (page  138.) 

^  =  rT7^-     •     •     •     •     IV.  (page  138.) 


Weight  of  air  : 
Constant  volume 


^  =  T4Si«^ V.(pagsm) 


New  pressure  =  p  (1  +  at) VI.  (page  141.) 

a  =  2T3  =  0.00367  for  perfect  gases. 

3  =  cTi^, VII.  (page  146.) 

B  for  air  =  39.272,  kilogrs.  Jc  =  1.41. 
Mariotte's  law  :    v,^j,  =  V2P2  =  v^p^,  etc.     .    VIII.  (page  147.) 
Mai'iotte  and  Gay  Lussac's  combined  : 

V,  Po  1     4-     at^  TV       ,  I  AC    ^ 

-^  —-i-^  .  -. 7^  ....     IX.  (page  148.) 

v^      p,      1  +  cxt^ 

D«  P„  1    +    at,  ^      ,  w  jn  N 

^P±  =  l^=-^&L,,U:.    .    XI.  (page  m, 

Ti  =»  273  +  t„  T.i=  273  +  t^,  etc. 
p,v,=BT„    p^v.^RT.,,    p,v^  =  RT3,  etc.     .    XII.  (page  151.) 

203 


204 

Isothermal  curve 


Isoclynamic  curve 


THEBMOD  TNAMIC8. 
pv  =PjV^  =  j}^^,  etc. 
i/=:3.3036i3wlog  — 

=  3.3026i),y,log—     . 

irx2.3026i9vlog^  .     . 

g  =  2.3026^i;wlog— .    . 

^  =  2.3026  ^i^r  log  — 


.  XIII.  (page  155.) 
,  XIV.  (page  155). 
.    XV.  (page  157). 


=  2.3026  ^iETi  log  ^. 

.     .    XVI.  (page  158.) 

Adiabatic  curve  : 

^.v^fc  =1^21-/^  =^3'y3^  etc.      . 

.    XVII.  (page  160.) 

Zf  =  -^  =  1.41  for  air. 

\vsj               pv      •     •     •     ' 

.    XVIII.  (page  162.) 

/vy-1       T,       273  +  ^1 
\v^)        ~  T  -  27d  +  t  • 

.     .    XIX.  (page  162.) 

k-1^  0.41  for  air  : 

^Kf)*  •  •  • 

.    .    XX.  (page  163.) 

-|  =  0.7093  for  air: 

(^)^-l  •  •  • 

.     .    XXI.  (page  163.) 

— ,—  =  0.2907  for  air  : 

L=-^{t,-t)r.^iT,-T). 


L 


^  =  abP 


-.-[-(f;)^^] 
.[-(-)'-■] 


XXII.  (page  165.) 
XXIII.  (page  166.) 
XXIVa.  (page  167.) 

XXIV&.  (page  167.) 


RECAPITULATION  OF  FOBMUL^.  205 

Transference  from  one  adiabatic  to  another  : 

f^  =  ^-^,etc.    .    .    .    XXV.  (page  175.) 

Expansion  under  constant  pressure  : 

L=p{v-v,)=pi{v-v^).     .    XXVI.  (page  176.) 
L  =  R{f  -  t^)^ R  {T -  T^)  .    XXVII.  (page  176.) 

Cycle  process  : 

^{Qi-Q2)  =  F.    .    .    XXVIII.  (page  181.) 

-^^(^1-^:)  =  --^.     .     .    XXIX.  (page  181.) 

F=^-^{T,-T.^ XXX.  (page  184.) 

F=-9j,^{T,-T,) XXXI.  (page  184.) 

F  =  2.3026  R{T,-  T.)  log  |-  .    XXXII.  (page  185.) 

F  z=  2.3026  R  (^1  -  t.,)  log  —   .    XXXIII.  (page  185.) 
Heat  weight : 

Al\  ~  Al\^  ~  ATi 
General  law  of  variation  of  pressure  with  volume : 

pm.^n  —  p^my^n  —  p^iny^n^  etc.    .     XXXIV.  (page  195.) 
/"^^YV^^^.    .     .     .    XXXV.  (page  196.) 

X=     ^    p,v,  ["l-  hh\'^~\    ,     .    XXXVI.  (page  197.) 

i  =  _^L_££zil(/,  _^    .    ,    .    .    XXXVII.  (page  197.) 

n  —  m       A 

Q  =  e{t-t,)  +  -^  c{k^l)  {t,  -  0.    XXXVIII.  (page  197.) 

Q^'^^dt-t,) XXXIX.  (page  198.) 

Q  =  s{t-~t,) XL,  (page  198.) 

s  =  ^^-'>^  c XLI.  (page  198.) 

m  —  n 


^J=.  =_%.=_%,  etc (page  187.) 


CHAPTER  VIII. 

COMPAEISON  OP  THE  HOT-AIR  ENGINE   AND   STEAM  ENGINE. — VAEIOUS 
KINDS   OF  HOT-AIR  ENGINES. 

Efficiency  of  the  Steam  Engine. — It  lias  been  proved  tliat  one 
unit  of  heat  is  equivalent  to  a  mechanical  work  of  424  meter- 
kilograms.  Now  one  kilogram  of  good  anthracite  furnishes 
about  7,500  heat  units,  that  is,  it  will  heat  7,500  kilograms  of 
water  one  degree,  or  will  raise  750  kilograms  of  water  10°.  If 
therefore,  all  the  heat  furnished  by  the  combustion  of  one  kilo- 
gram of  coal  could  be  transformed  into  mechanical  work,  we 
should  have  7,500  x  424  =  3,180,000  meter-kilograms.  This 
work  would  be  obtained  in  one  second  if  the  combustion  occu- 
pied one  second,  in  one  hour  if  the  combustion  occupied  one 

hour.     In  the  latter  case,  the  delivery  would  be  —  =  883 

obOU 

883 
meter-kilograms  per  second,  or  -^^-^  =  about  12  horse-power. 

Experiments  upon  the  steam  engine  have  shown  that,  even  in 
the  best,  about  2  kilograms  of  coal  are  necessary  for  one  horse- 
power, and  therefore  only  ^  of  the  work  in  the  fuel  is  ob- 
tained.    Most  engines  use  from  1  ^  to  2  times  as  much  coal,  so 

A 

that  their  efficiency  is  only  24^^1:5=36    °^    24'^2"48" 

At  first  sight  it  would  seem  that  the  reason  of  this  is  to  be 
sought  in  defective  boiler  construction,  setting,  etc.  But  in  all 
these  respects  but  little  room  for  improvement  now  remains. 
Hence  Redtenbacher  concluded  that  better  results  could  only 
be  obtained  by  an  entire  change  in  the  method  of  conversion 
of  heat  into  work.  What  sort  of  change  is  necessary,  he  has 
not  informed  us.     In  the  present  condition  of  science  it  would 

206 


WOBK  OF  ONE  KILOGRAM  OF  WATEB. 


207 


be  hard  to  find  any  one  competent  to  give  us  sucli  information. 
Meanwhile  Zeuner  has  shown  that  our  best  steam  engines,  as 
well  as  hot-air  engines,  utilize  the  fuel  exceedingly  well,  and 
that  we  demand  an  impossibility  when  we  require  these  ma- 
chines to  utilize  all  or  even  the  greatest  part  of  the  heat  con- 
tained in  the  fuel.  It'would  be  as  reasonable  to  expect  a  water 
wheel  to  utilize  the  entire  fall,  from  the  source  to  the  sea,  of 
the  river  which  moves  it. 

Still,  the  views  held  as  to  the  imperfect  utilization  of  the 
fuel  by  the  steam  engine,  and  perhaps  also  the  danger  of 
explosions,  which  unfortunately  still  remains,  led  to  the  con- 
struction of  the  hot-air  engine. 

We  know  that  water  requires  to  convert  it  into  steam  about 
540  heat  units.  This  enormous  amount  of  heat  is  required 
simply  to  convert  the  liquid  water  into  a  gas.  Since  the  air  is 
already  a  gas,  no  heat  is  needed  for  such  a  transformation,  and 
hence  it  would  seem  to  be  much  cheaper  than  steam.  This 
however  is  not  the  case,  as  will  be  seen  hereafter. 


Work  wJiicJi  One  Kilogram  of  Water  Performs  in  Evaporation. — 
Let  ABCD  be  a  cylinder  whose  cross-section  is  exactly  one 
square  meter,  Fig.  28.  In  the  bottom  is  one  cubic 
decimeter,  or  one  kilogram,  of  water  at  0^.  Upon 
the  surface  of  the  water  rests  the  air-tight  piston 
KK.  Since  the  cross-section  of  the  cylinder  is 
one  square  meter,  the  depth  of  the  water  is  one 
millimeter ;  for  yoVt  of  ^  cubic  meter  =  1  cubic 
decimeter. 

Suppose  the  piston  KK  loaded  with  10,334  kilo- 
grams, and  that  there  is  a  vacuum  above  it.  The 
pressure  of  10,334  kilograms  corresponds  then  to 
that  of  the  atmosphere  at  0°  and  TGO'""'-  of  ba- 
rometer. 

If  now  we  heat  the  water  up  to  100",  any  fur- 
ther addition  of  heat  generates  steam  of  10,334 
kilograms  pressure.  It  is  known  to  be  a  physical 
fact,  to  which  we  shall  return  when  we  come  to  KilOiEilK 
speak  of  steam  and  the  steam  engine,  that  steam 
generation  will  not  commence  until  the  tempera- 
ture of  100""  is  attained.     If  it  is  desired  to  make 


D 


208  THEBM0DYNAMIC8. 

it  occur  earlier,  the  pressure  on  the  piston  must  be  diminished. 
The  more  heat  we  impart,  the  more  steam  is  generated  and  the 
piston  is  raised  ever  higher,  while  the  temperature  remains  at 
lOO"".  When  all  the  water  is  converted  into  steam,  the  piston 
will  be  about  1734:  millimeters  or  1.734  meters  above  the  bot- 
tom, since  the  one  cubic  decimeter  of  water  will  give  about  1734 
cubic  decimeters  of  steam.  The  pressure  10,334  kilograms  has 
thus  been  raised  1.734  meters,  which  corresponds  to  a  me- 
chanical work  of  17,919  meter-kilograms. 

Since  we  must  impart  about  540  heat  units  to  change  the 
water  at  100^  into  steam  at  100°,  and  also  100  heat  units  to 
raise  the  water  from  0°  to  100°,  we  have  imparted  altogether 
about  640  heat  units  in  obtaining  the  above  work. 

If  the  piston  is  loaded  with  2  x  10334  kilograms,  steam  gen- 
eration commences  at  121°,  and  the  water  must  be  heated  to 
this  temperature  before  the  piston  is  raised.  If  all  the  water 
is  converted  into  steam,  the  height  to  which  KK  is  raised  is 
914  millimeters  =  0.914  meters,  and  the  one  cubic  decimeter 
of  water  furnishes  914  cubic  decimeters  of  steam  of  2  atmos- 
pheres pressure.  Since  the  steam  occupies  a  space  of  914 
cubic  decimeters  while  before  it  occupied  1734,  we  see  that  its 
density  is  nearly  double  as  great  as  before. 

But  now  the  quantity  of  heat  imparted  in  this  case,  in  order 
to  completely  vaporize  the  water  is  only  about  640  heat  units. 
The  work  performed  however  is  2  x  10334  x  0.914  =  18890 
meter-kilograms,  or  greater  than  before. 

If  we  load  the  piston  with  3  x  10334  kilograms,  vaporization 
takes  place  at  about  135°.  The  steam  occupies  the  space  of 
620  cubic  decimeters,  and  the  piston  is  raised  through  620 
millimeters.  The  work  performed  is  3  x  10334  x  0.621  =  19252 
meter-kilograms,  while  the  heat  imparted  is  still  only  about 
about  640  heat  units,  and  so  on. 

We  see  that  the  work  of  the  steam  is  greater  the  more  the 
piston  is  loaded,  that  is,  the  more  the  water  is  heated,  or  the 
greater  the  expansive  force. 

In  all  cases,  however,  the  heat  required  for  the  outer  work 
is  much  less  than  that  required  for  the  vaporization.  In  the 
1st  case  the  latter  is  540  x  424  =  228960  meter-kilograms ;  in 
the  2d,  (640  - 121)  424  =  220056,  and  in  the  3d,  (640  - 135) 
424  =  214120  meter-kilograms. 


HISTORICAL  NOTE   UPON  HOT-AIR  ENGINES.  209 

This  circumstance,  that  most  of  the  heat  serves  only  to  va- 
porize the  water — to  separate  the  molecules — has,  as  we  have 
remarked,  led  to  the  idea  of  using  some  naturally  gaseous 
body,  and  of  these  there  are  none  more  suitable  than  the  air. 

Historical  Note  upon  Hot-Air  Engines. — The  first  to  apply  the 
idea  of  using  hot  air  appears  to  have  been  John  Stirling,  of 
Glasgow.  In  the  year  1827  he  devised  an  air  engine,  the  con- 
struction and  efficiency  of  which  are  not  now  known.  Six  years 
later  John  Ericsson  constructed  a  similar  engine  in  London, 
which  he  called  a  ^'caloric  engine^  The  invention  made  little 
progress  in  England,  although  such  men  as  Faraday  and  Ure 
were  interested  in  it,  and  Ericsson  removed  to  America,  where 
his  activity  in  many  directions  has  been  so  marked.  Here  he 
worked  at  the  perfecting  of  his  caloric  engine,  and  in  1848  he 
succeeded  in  introducing  his  first  engine  on  an  improved  sys- 
tem, in  the  Delamater  iron  foundry  in  New  York.  It  was  only 
5  horse  power,  but  in  the  following  year  one  of  60  horse  power 
was  set  up,  and  in  1851  a  caloric  engine  was  exhibited  at  the 
London  exposition.  The  engine  thus  became  more  widely 
known,  but  was  regarded  in  Europe  more  as  an  interesting  toy, 
incapable  of  competing  in  practice  Avith  the  steam  engine. 

This  view  was  by  no  means  contradicted  by  the  experiments 
made  in  America,  in  the  following  years.  The  invention  of 
Ericsson  was  followed  up  with  great  zeal,  and  a  company  was 
formed  with  John  B.  Kitching  at  its  head,  to  which  also  the 
Secretary  of  the  Navy,  Kennedy,  belonged,  to  build  a  large 
vessel  with  caloric  engines  called  the  "Ericsson."  This  vessel 
made  its  first  voyage  on  the  15th  of  February,  1853. 

The  vessel  was  2200  tons,  and  had  four  caloric  engines, 
which  set  in  motion  two  paddle-wheels.  The  cylinder  was  14 
feet  in  diameter  and  6  feet  stroke,  and  the  air  was  heated  to 
195°  C.  In  order  to  prevent  loss  of  heat  and  reduce  the  ex- 
penditure of  fuel,  there  were  four  remarkable  contrivances  near 
the  cylinder,  called  "regenerators."  Each  of  these  consisted 
of  a  network  of  wires,  whose  combined  length  was  nearly  fifty 
miles.  Before  the  air  escaped  it  passed  through  these  re- 
generators, and  gave  up  its  heat  to  the  wires,  which  then  were 
ready  to  impart  heat  to  the  fresh  charge  of  air.  The  engine, 
according  to  Ericsson's  calculation,  was  600  horse  power,  and 
14 


210  THERMODTNAMICS. 

since  it  consumed  in  twenty-four  liours  eight  tons  of  coal,  eacli 
horse  power  per  hour  required  the  extraordinarily  small  quan- 
tity of  1.11  pounds,  or  about  i  kilogram.  According  to  the  re- 
port of  Prof.  Wm.  A.  Norton,*  however,  the  power  of  the  engine 
on  the  trial  trip  was  only  300  horse  power,  which  gives  2.2 
pounds  of  coal  per  horse  power  per  hour,  an  amount  which, 
in  comparison  with  the  best  marine  engines,  which  require 
about  3.11  pounds,  shows  still  a  noticeable  economy-. 

Although  as  regards  economy  of  fuel,  therefore,  the  engine 
was  preferable  to  an  ordinary  steam  engine,  it  had  the  dis- 
advantage of  much  greater  weight  and  space.  Prof.  Norton 
came  therefore  to  the  conclusion,  that,  in  its  present  condition, 
the  engine  was  not  suited  for  marine  use  or  for  locomotives, 
but  that  where  weight  was  of  less  account,  and  economy  of  fuel 
was  desired,  it  possessed  many  good  qualities. 

Ericsson  himself  noticed  during  the  test  many  defects,  which 
upon  his  return  to  New  York  he  sought  to  remedy.  Instead 
of  four  cylinders  he  used  two  of  somewhat  less  diameter,  but 
longer  stroke.  But  the  test  with  this  new  apparatus  gave 
more  unfavorable  results  than  the  first,  and  so  in  the  begin- 
ning of  1854,  the  engines  were  taken  out  and  replaced  by  ordi- 
nary marine  steam  engines. 

Ericsson  now  busied  himself  with  the  construction  of  smaller 
machines,  for  the  purposes  of  the  lesser  industries.  In  1860 
he  succeeded  in  producing  an  engine  which  found  general  ac- 
ceptance not  only  in  America,  but  also  in  France,  Germany, 
and  Sweden.  In  Germany  the  Director  of  the  workshops  of 
the  Hamburg  Magdeburg  Steamship  Company,  Andrea,  spe- 
cially interested  himself  in  the  introduction  of  the  new  inven- 
tion. But  it  also  has  not  realized  the  hopes  which  were  placed 
in  it.  After  a  few  years  it  was  removed  from  many  establish- 
ments, and  replaced  by  the  steam  engine.  We  shall  have  oc- 
casion later  on  to  describe  the  Ericsson  engine  in  detail. 

Comparison  of  the  Work  jjerformed  hy  Hot  Air  and  Steam. — 
Let  us  see  now  whether  hot  air  is  in  fact  a  cheaper  motor  than 
steam. 

We  suppose  again  under  the  piston  KK,  Fig.  29,  whose  cross- 
section  is   one  square  meter,  one  kilogram  of  air  at  0°  and 

*  American  Journal  of  Sciences  and  Arts,  Sd  Series,  vol.  xv.,  May,  1853. 


WOBK  DTJRmG  EXPANSION  OF  AIR. 


211 


atmosplieric  pressure.  Tliis  occupies  a  space  of  0.7733  cubic 
meters,  and  hence  the  piston  KK  is  at  a  distance  0.7733 
meters  from  the  bottom.     Let  there  be  a  vacuum    a  u 

above  the  piston,  and  let  it  be  loaded  with  10,334      "  ' 

kilograms. 

The  air  cannot  expand  because  its  pressure  is 
in  equilibrium  with  the  piston  pressure. 

If  now  we  heat  the  inclosed  air  graduallj,  up  to 
273°,  it  expands,  as  we  know,  to  double  its  volume, 
and  the  piston  is  raised  0.7733  meters.  The  work 
performed  is  hence  10334  x  0.7733  =  7991  meter- 
kilograms,  and  the  heat  imparted  is  0.2375  x  273 
=  64.84  heat  units. 

When  we  converted  1  kilogram  of  water   into 
steam  of  one  atmosphere,  we  imparted  640  heat 
units,  and  obtained  a  work  of  17,919  meter-kilo- 
grams.    For  a  work  of  only  7,791  kilograms,  we 
should  need  to  evaporate  only  ttViV  =  O-'^S  kilo-  ^ 
grams  of  water,  for  which  we  should  impart  only 
640  X  0.45  =  288  heat  units.    We  obtain,  therefore,  D 
in  fact,  by  means  of  air,  as  much  work  by  the  ex- 
penditure of  64.94  heat  units  as  by  the  expenditure  of  288  heat 
units  with  water ;  or  inversely,  if  we  wish  to  obtain  the  same 

288 
work  with  water  as  with  air,  we  must  use  „  .  ^  .  =  4.44  times  as 

64.  o4 

much  fuel. 

But  now,  by  the  use  of  steam  we  can  cause  a  good  vacuum 
above  the  piston.  Thus,  in  a  condensing  engine,  the  back  press- 
ure upon  the  piston  is  only  ^  to  4^  of  an  atmosphere.  In  our 
illustration  with  steam  then,  already  discussed,  after  the  pis- 
ton has  moved  through  1.734  meters,  we  can  gradually  remove 
the  pressure  from  it  down  to  |  x  10334  or  |  x  10334  kilograms, 
during  wliich  the  steam  expands,  performing  work  at  the  ex- 
pense of  its  inner  work,  and  thus  a  considerable  amount  of 
mechanical  work  can  be  obtained  without  further  expenditure 
of  heat.  In  the  hot-air  engine,  on  the  other  hand,"  we  cannot 
make  the  back  pressure  less  than  one  atmosphere.  On  this 
account  the  difference  in  fuel  consumption  will  be  less  than  as 
above  computed. 

To  this  we  may  add  that  with  steam  the  engine  can  be  much 


212  THEBMODYNAMICS. 

smaller  than  with  air.  For  with  steam,  as  we  have  seen,  for  a 
work  of  7,991  kilograms  we  need  to  vaporize  only  0.45  kilograms 
of  water.  The  piston  then  will  be  raised  only  1.734  x  0.45  = 
0.7803  meters,  and  the  entire  space  required  is  0.7813  cubic 
meters.  With  air,  on  the  contrary,  we  require  2  x  0.7733  = 
1.5466  cubic  meters,  or  almost  twice  as  much  space.  (It  is 
worth  remarking  that  with  the  steam  engine  we  must  keep  on 
hand  20  or  30  times  as  much  steam  as  is  used  per  stroke,  so 
that  the  entire  apparatus,  boiler  and  all,  may  occupy  more  space 
than  the  hot-air  engine.) 

If  we  compress  the  air  under  the  piston  to  3  atmospheres,  it 
occupies  only  ^d.  of  its  volume  for  one  atmosphere.  We  must 
then  load  the  piston  with  3  x  10334  kilograms.     Its  height  will 

0.7733 

then  be  — ^^ —  —  0.2578  meters.     Upon  heating  to  273"*  it  rises 
o 

0.7733 
to  double  this  height,  and  the  work  done  is  3  x  10334  x  -^-- — 

o 

^  10334  X  0.7733  =  7991  meter-kilograms,  while,  as  before,  64.8 
heat-units  are  required.  By  gradually  diminishing  the  press- 
ure upon  the  piston  down  to  10,334  kilograms,  we  can  now 
obtain  work  by  the  expansion  of  the  air.  The  work  thus 
obtained  is,  however,  in  part  lost  when  the  air  is  again  com- 
pressed to  three  atmospheres,  and  thus  brought  back  to  its 
original  condition. 

Let  us  compare  with  this  the  example  already  given,  where 
1  kilogram  of  water  was  converted  into  steam  of  3  atmospheres. 
If  here  we  wish  a  work  of  only  7,991  meter-kilograms,  we  have 

7791 
to  vaporize  only      „      =  0.405  kilograms  of  water,  and  there- 
19252 

259  2 
fore  expend  640  x  0.405  =  259.2  heat-units,  or  still   -prrr^  -■  ^ 
^  64.8 

times  as  much  as  for  the  air.  But  now,  by  expansion  down  to 
^th  to  ^th  of  an  atmosphere,  we  can  obtain  considerable  work, 
almost  none  of  which  is  lost,  because  the  water  mass  of  0.405 
kilograms  which  we  have  to  force  back  into  the  boiler,  occupies 
a  very  small  space,  so  that  the  work  required  to  force  it  in  is 
very  small.  For  such  reasons,  then,  scarcely  double  as  much 
fuel  may  be  required  as  for  air. 

From  the  above  it  follows,  that  hot  air  is,  in  general,  a  cheaper 
motor  than  steam,  but  the  difference  is  less  the  greater  the 


EBICSSOJ^  HOT-AIR  ENGINE. 


21^ 


tension  of  the  steam  and  tlie  greater  the  expansion.  To  this 
we  must  acid  that  the  engine  proper  for  the  same  power  is  less 
in  size  for  the  steam  than  air,  and  that  the  steam  requires  to  be 
heated  to  a  much  less  temperature  than  the  air.  This  allows 
the  steam  piston  and  stuffing  boxes  to  be  better  lubricated,  and 
materially  reduces  the  friction.  On  the  other  hand,  the  hot-air 
engine  is  free  from  danger  of  explosion,  and  requires  little  or 
no  water. 

Let  us  now  examine  the  construction  and  theory  of  these  en- 
gines a  little  more  closely. 


I.  OPEN  HOT-AIR  ENGINE  WITH  OPEN  FIREPLACE,  IN  WHICH 
THE  HOT  AIR  IS  EXPELLED  AT  EACH  STROKE.  CALORIC  EN- 
GINE OP  ERICSSON— SYSTEM  OF  1860. 


In  Fig.  30  we  haye  an  ideal  section  of  such  an  engine.  The 
cylinder  is  abed.  In  the  right  half  of  it  are  two  pistons  pq  and 
rs,  of  which  one  is  the 
working  piston  and  the  l\, 

other  the  feed  piston. 

In  the  back  part  of 
the  cylinder  is  an  iron 
fire-box  efgli;  ih  is  the 
grate,  I  the  ash-pit.  The 
hot  gases  pass  from  the 
fire  space  through  the 
pipe  mn  into  the  an- 
.  nular  space  viv  which 
surrounds  the  cylinder 
cibcd.  In  this  way  as 
much  heat  as  possible 

is  imparted  to  the  cylinder.     The  hot  air  escapes  from  mo  by 
the  chimney  oo  into  the  outer  air. 

The  working  piston ^^'g'  li^^  valves,  two  of  which  are  shown  at 
t  and  u.  These  valves  open  toward  the  left.  The  motion  of  this 
piston  is  transferred,  by  means  of  two  rods  and  lever  work,  to  & 
fly-wheel,  one  half  of  which  is  heavier  than  the  other.  From  this, 
by  means  of  mechanism,  motion  is  imparted  to  the  feed  piston 
Ts,  so  that  it  has  a  greater  velocity  than  the  working  piston. 


214  THERMODYNAMICS. 

To  the  left  of  tlie  feed  piston  is  fastened  a  cylinder,  zz,  of 
tliin  cast-steel  plate,  which  has  a  somewhat  greater  diameter 
than  the  fire-box.  The  fire-box  is  surrounded  by  another  cyl- 
inder of  cast-steel,  xy,  which  receives  the  heat  radiated  from 
the  cylinder  and  fire-box. 

Finally  we  have  at  A,  a  pipe  with  a  valve  d,  through  which 
the  hot  air  escapes  after  acting  in  the  engine.  As  A  commu- 
nicates with  the  chimney  oo,  the  hot  air  and  products  of  com- 
bustion are  discharged  together  into  the  outer  air,  or  may  be 
discharged  into  a  closed  space. 

Method  of  Action. — Let  us  now  consider  the  method  of  action 
of  the  engine.  First,  the  fly-wheel  is  turned  by  a  simple  appa- 
ratus, so  that  the  centre  of  gravity  of  the  heavy  half  lies  a  little 
to  one  side  of  the  highest  point.  The  centre  of  gravity  then 
sinks  of  itself,  and  the  fly-wheel  turns  180^.  This  motion  is 
imparted  by  means  of  link  work,  to  the  pistons  pq  and  rs.  But 
since  the  feed  piston  rs  moves  more  rapidly  than  the  working 
piston,  pq,  a  partial  vacuum  is  caused  between.  In  conse- 
quence of  this,  the  valves  t  and  u  in  the  working  piston  open, 
and  air  enters  between  the  pistons.  The  valve  a,  in  the  feed 
piston,  which  possesses  a  very  different  form  from  that  shown  in 
the  Figure,  and  is  applied  at  a  different  place — closes.  The  hot 
air  remaining  in  the  cylinder  departs  through  d,  which  remains 
open  during  the  entire  motion  of  the  feed  piston  from  right  to 
left. 

When  the  feed  piston  has  reached  its  extreme  position  on 
the  left,  the  working  piston  has  not  completed  its  stroke,  but 
still  moves  toward  the  left,  while  the  feed  piston  now  moves 
toward  the  right.  The  air  between  is  thus  compressed,  and 
the  valve  a  opens,  and  admits  a  portion  of  the  air  into  the  hot 
space  left  of  the  feed  piston. 

This  air,  which  enters  cold,  is  now  rapidly  heated,  especially 
by  the  hot  plates  xx  and  zz,  and  thus  has  a  greater  expansive 
force.  This  pressure  is  distributed  according  to  the  law  of 
distribution  of  pressure  in  a  fluid,  over  the  entire  volume  of 
air,  that  is,  it  acts  not  only  upon  the  feed  piston,  but  also  upon 
the  working  piston,  which  it  forces  toward  the  right. 

But  when  this  motion  towards  the  right  begins,  the  feed  pis- 
ton is  already  traveling  in  the  same  direction,  and  since  it 


EBICSS'OW  HOT-AIR  ENGINE.  215 

moves  faster  than  tlie  working  piston,  more  cold  air  is  being 
continually  forced  through  a  into  the  hot  space.  The  tension 
of  the  inclosed  air  thus  increases,  and  reaches  its  maximum 
when  the  feed  piston  has  passed  through  half  its  stroke,  and 
has  its  greatest  velocity. 

From  this  point  it  diminishes  gradually  down  to  about  1.16 
atmospheres,  when  the  feed  piston  is  at  the  end  of  its  stroke. 

At  this  moment  the  exhaust  valve  d  opens,  and  the  hot  air 
escapes  rapidly,  while  the  feed  piston  returns  towards  the 
left.  While  then  the  tension  of  the  hot  air  is  sinking  to  one 
atmosphere,  the  working  piston  returns  a  certain  amount,  about 
T^th  of  its  entire  stroke,  so  that  we  can  assume  during  this 
period  the  pressure'  equal  upon  both  sides  of  the  working  pis- 
ton. 

It  should  be  especially  remarked,  that  the  valve  a,  during 
forward  motion  of  the  feed  piston,  remains  constantly  open, 
that,  therefore,  the  increased  tension  of  the  hot  air  is  trans- 
ferred to  the  cold  air  between  the  pistons,  and  that  hence,  only 
the  working  piston  is  impelled  by  the  increased  tension.  For 
this  reason  it  is  called  the  "  icorJcing  piston."  The  rear  piston, 
by  means  of  its  greater  velocity,  forces  the  cold  air  into  the 
hot  space,  and  is  hence  called  the  "feed  piston."  By  this  ar- 
rangement the  working  piston  is  shielded  from  the  radiant 
heat  of  the  hot  portion  of  the  cylinder,  and  can  be  lubricated 
and  kept  in  good  condition.  In  fact,  in  this  lies  a  great  part 
of  the.  ingenuity  of  the  whole  invention.  During  the  back- 
ward motion  of  the  feed  piston,  the  valve  a  is  closed. 

In  order  that  the  cold  air,  when  it  arrives  in  the  hot  space 
behind  the  feed  piston  may  be  heated  quickly,  it  must  enter 
more  readily  than  the  valve  a  would 
allow,  and  the  following  arrangement  ^ 
is  adopted.  Upon  the  circumference  S 
of  the  feed  piston  rs,  rectangular 
notches  are  cut,  from  I  to  1  inch 
broad,  by  i  deep.  In  the  Figure  31 
only  two  can  be  seen  at  oo.  Behind 
the  piston  is  a  steel  ring,  tf,  which 
fits  the  cylinder  snug,  but  whose  inner 

circumference  is  about  I  inch  from  the  feed  j)iston.  Thus  in 
the   position   shown   in   the   Figure,    communication   is   open 


216 


THERMOD  TNAMIGS. 


between  tlie  space  right  and  left  of  tlie  feed  piston,  and  cold 
air  can  pass  tlirongli.  The  ring  is  held  in  place  on  the  left  by 
several  pins,  upon  the  circumference  of  the  back  extension  of 
the  piston.  When  the  piston  moves  toward  the  left  there  is 
a  partial  vacuum  between  the  pistons,  and  one  atmosphere 
pressure  on  the  left,  hence  the  steel  ring  is  pressed  close  up 
to  the  piston  face,  and  the  holes  oo  are  covered  by  it  and 
closed. 

In  order  to  make  the  action  of  the  machine  still  clearer,  we 
have  given  in  Figures  32,  33,  34, 
35  and  36,  the  principal  positions 
of  the  pistons. 

In  Fig.  32  the  working  piston 
has  arrived  at  the  end  of  its  for- 
ward stroke,  and  the  feed  piston 
is  already  started  on  the  back 
stroke.  The  tension  of  the  air  in 
the  engine  is  one  atmosphere,  both  between  the  pistons  and 
back  of  the  feed  piston. 

In  Fig.  33  the  feed  piston  is  at 
the  end  of  its  back  stroke,  while 
the  working  piston  has  moved 
only  a  part  of  its  way  toward  the 
left.  Between  the  two  is  cold  air 
of  one  atmosphere  tension. 

In  Fig.  34  the  feed  piston  has 
commenced  its  forward  stroke  while  the  working  piston  has 
arrived  at  the  left  end  of  its  stroke.     The  air  between  is  com- 
pressed, the  valve  in  the  feed  pis- 
ton is  open,  and  in  the   working 
piston  shut,  and  cold  air  is  passing 
into  the  hot  space  behind.     This 
air  when  heated  communicates  its 
higher  pressure,  according  to  the 
laws  of  fluid  pressure,  through  the 
entire  volume  of  air,  and  thus  acts 
upon  the  working  piston. 

In  Fig.  35  the  feed  piston  is  at  the  middle  of  its  forward 
stroke,  where  it  has  its  greatest  velocity,  and  where  the  greater 
part  of  the  air  has  been  forced  through  and  heated.     Here 


EBICSSOIi  HOT- AIM  ENGINE. 


217 


tlien  must  be  tlie  maximum  tension.  Botli  pistons  now  travel 
nearly  together,  and  but  little  more  air  enters  the  heated  space, 
so  that  now  the  pressure  diminishes  as  the  volume  increases, 
and  the  air  works  exjoansively. 

Finally,  in  Fig.  36  the  feed  pis- 
ton has  reached  the  end  of  its  for- 
ward stroke.  At  this  moment  the 
exhaust  valve  opens,  and  the  ten- 
sion of  the  air  falls  to  one  atmos- 
phere.     The  working  piston  has  ^^^-  ^^• 

still  a  portion  of  its  stroke  to  go, 
while  the  feed  piston  moves  back. 
"When  the  working  piston  arrives 
at  the  end  of  its  stroke  we  have 
the  position  of  Fig.  32. 

Fig.  86.  Variation   of  Pressure. — Let   us 

now  determine  the  air  tension  at  the  various  positions  indi- 
cated in  Figs.  32,  33,  34,  35  and  36. 

In  Figs.  32,  33,  the  air  right  and  left  of  the  feed  piston  has 
the  pressure  of  the  atmosphere,  which  we  denote  by  p.  Let  the 
volume  of  air  in  the  space  r^  be  A  cubic  meters,  and  let  its 
absolute  temperature  be  T^°.  Behind  the  feed  piston  let  there 
be  confined  B  cubic  meters  at  T^°. 

If  now  the  specific  volume  (volume  of  one  kilogram)  of  the 
A  cubic  meters  be  Va,  and  that  of  B  be  v^,  then  we  have  from 
Equation  XII., 

P 
and 

P 


]^a  =  RT^     or 


P'Vb 


RT^    or    V, 


(1). 


(2). 


If  we  denote  the  weight  of  the  A  cubic  meters  by  G^,  and 
that  of  B  by  G^,  we  have 

^"  =  .- •     (3). 


G,^ 


Vb 


(4). 


218  THEBM0BYNAM1C8. 

If  we  insert  in  (3j  and  (4)  the  values  of  v,^  and  v^,  given  by  (1) 
and  (2)  we  liave 

^  TVT ^'' 

ft  =  |f^   .......     (6,. 

Therefore,  from  (5)  we  can  find  the  weight  of  cold  air  in- 
closed, and  from  {^)  that  of  the  warm  air.  The  entire  weight 
of  air  is  then 

'p  (  A       B 


G=G,  +  G,^^[^-^^+^y    .     .    .    (7). 

and  this  is  the  weight  of  air  contained  in  the  engine  for  all  the 
other  positions  during  the  forward  feed  stroke. 

In  the  space  rg.  Fig.  34,  let  there  be  C  cubic  meters  of  air  of 
the  temperature  T^°,  and  in  m  D  cubic  meters  with  the  tem- 
perature ^2°,  and  let  the  pressure  be  px.  We  find  as  in  (5)  for 
the  weight  of  G, 

Gp^ 

and  for  the  weight  of  D, 

Dp, 
BT,' 

Both  weights  must  together  be  equal  to  G,  hence 

^.   /  n-      Ji\ 

(8). 


G-- 

^Pi  fG- 

*«) 

and 

putting  (7) 

and 

(8)  equal  and 

reducing 

p 

AT,  + 

GT,  + 

DT, 

From  this  we  can  determine  the  ratio  of  the  pressures  for 
the  position  in  Fig.  33  to  that  in  Fig,  34. 

In  like  manner,  in  Fig.  35  let  us  have  in  r^,  E  cubic  meters  at 

T^",  and  in  n,  H  cubic  meters  at  Ti°,  and  the  tension  in  both 

E'o 
spaces  jjo.     Then  we  have  for  the  weight  in  E  as  before  j^y^ 


UBICSSOJ!^  HOT-AIB  ENGmE.  219 


and  for  tlie  weight  of  H,  j^y; ,  and  hence 


From  (10)  and  (7) 

p  "  ET,  +  HT, ^^^^■ 

This  gives  then  the  maximum  tension  of  the  air. 

If  in  Fig.  36  the  volume  in  r^  is  /cubic  meters  and  that  in  o, 
K  cubic  meters,  and  if  the  tension  in  both  spaces  is  p^,  we  shall 
have  in  similar  manner, 

p,_AT,  +  BT, 

Let  now  the  cross-section  of  the  cylinder  be  i^  square  meters, 
and  the  distance  between  the  pistons  in  Fig.  33,  be  r^,  in  Fig. 
34,  rg,  in  Fig.  35,  r^,  in  Fig.  36,  r^ ;  then 

A  =  Fr^,     C^  Frs,     E  =  Fn,     and    /=  Fo^^- 

If  also,  instead  of  the  annular  space  behind  the  feed  piston 
in  Fig.  33,  we  suppose  a  cylindrical  space  of  equal  volume,  whose 
length  is  b,  we  have 

B  =  Fb    and    h  =  4^. 
F 

If  in  Fig.  34  the  distance  of  the  feed  piston  from  the  fire-box 
is  TO,  in  Fig.  35,  n,  in  Fig.  36,  o,  we  can  put 

B  =  Fb;  B^Fm  +Fb  =  (6  +  to)  F;  H=  (b  +  n)F 
K^  {b  +  o)F. 

Substituting  these  values  in  (7),  (9),  (11),  and  (12),  we  obtain 
the  following  formulae : 


G  = 

.  Fp 
R 

(t 

*i) 

Pi  _ 

^'; 

.^2   + 

bT, 

P 

r,T, 

+  (6 

+  to)  1\ 

(a). 
(5). 


i^2_ 

r.T,  +  hT, 

V 

nT.,+  {lj  +  n)'l\ 

Pz  _ 

nT-i  +  hT^ 

p 

r,T,  +  {b  +  o)T^ 

220  THERMODYNAMICS. 

(c). 

{d). 

According  to  Boetius  we  have  for  a  one  liorse-power  caloric 
engine, 

0  —  stroke  of  feed  piston  =  0.418  meters. 

1  =  stroke  of  working  piston  =  0.22  meters. 

r^  =  0.275  meters,     m  =  0.054 

rs  =  0.180      "  w  =  0.209 

r4  =  0.063      "  n  =  0.025 

F  —  0.165  square  meters. 

Further,  B  —  Fb  —  0.2  of  the  entire  space  Fo. 

Hence 

Fb  =  0.2  X  0.165  X  0.418  =  0.013794  cubic  meters, 
and 

b  =  0.0836  meters. 

If  we  assume  that  the  cold  air  between  the  pistons  preserves 
in  all  positions  the  constant  absolute  temperature  T^  =  273  +  10 
—  283°,  and  the  hot  air  behind  the  feed  piston  has  the  temper- 
ature always  of  T.  =  273  +  300  =  573°,  we  have  from  (a),  ^(6), 
(c),  etc.,  the  following  results  : 

_  0.165  X  10334  /0.275      0.0836\ 
29.272        V  283    "^    573  / 
or 

G  =  0.06524  kilograms. 

The  weight  of  cold  air  drawn  in  at  each  stroke  is 

^         Ap       Fpr,      0.165  x  10334  x  0.275      ^  ^^.-  ,  ., 

^^^  =-m=  BT,  ^ 29:272^^183 =  ^"^^^^  kilograms. 

Since  the  machine  made  45  revolutions  in  one  minute,  the 

45 

weight  of  air  used  per  second  is  ^  x  0.0565  =  0.0424  kilograms. 

Also  from  (Z>)  we  obtain 

£i  ^ 0.275  X  573  +  0.0836  x  283 ^  181.064  _  ^  ^  . 

p  ~  0.180  X  573  +  (0.0836  +  0.054)  283  ~  141.911  ~ 


EBIG8S0N  HOT-AIB  ENGINE.  221 

From  (c) 

p,_ 181.064 181.064  _ 

p  ~  0.063  X  573  +  (0.0836  +  0.209)  283  "  118.735  ~ 

From  {d) 

P3_ 181.064 _18L064_ 

p  ~  0.025  X  573  +  (0.0836  +  0.418)  283  ~  156.108  ~ 

Let  us  now  find  the  delivery  of  tlie  engine  under  the  assump- 
tion, which  is  in  fact  very  nearly  correct,  that  the  tension  varies 
as  the  ordinates  to  a  straight  line. 

In  Fig.  34,  the  working  piston  has  its  extreme  position  on  the 
left.  The  air  behind  it  has  a  tension  of  1.276  atmospheres,  or 
is  0.276  atmospheres  in  excess  of  the  outer-air  pressure. 

In  Fig.  35,  the  maximum  tension  is  1.525,  or  0.525  atmos- 
pheres in  excess  of  the  outer  air. 

The  mean  effective  pressure  upon  the  working  piston,  while 
passing  from  the  position  in  Fig.  34  to  that  in  Fig.  35,  is,  if  the 

pressure  varies  as  the  ordinates  to  a  straight  line,  ~ ^ — '- 

=  0.4005  atmospheres.  Since  the  distance  passed  through  by 
the  working  piston  is  0.038  meters,  we  have  for  the  work  per- 
formed 

0.4005  X  10334  x  0.165  x  0.038  =  25.983  meter-kilograms. 

Since  the  effective  pressure  in  Fig.  36,  is  1.16  —  1  =  0.160 
atmospheres,  we  have  for  the  mean  pressure,  while  the  work- 
ing piston  moves  from  the  position  in  Fig.  35  to  that  in  Fig. 

36,  ~ ^ — '- =  0.343  atmospheres.     The  distance  passed 

through  is  0.172  meters,  and  hence  the  work  done  is 

0.343  X  10334  x  0.165  x  0.172  =  100.594  meter-kilograms. 

The  total  work  is  therefore 

25.983  +  100.594  =  126.577  meter-kilograms. 

From  this  we  must  sub  bract  the  work  done  in  compressinpj 
the  air,  which  compression  commences  at  the  position  of  Fig, 
33  and  lasts  to  Fig.  34.     The  final  pressure  is  0.276  effective. 


222  THERMODYNAMICS. 

and  hence  tlie  mean  pressure  is  ^-^ — ,  and  tlie  work  done  is 

^^  X  10334  X  0.165  X  0.041  =  9.668  meter-kilograms. 

Hence  the  effective  work  done  per  revolution  is 

126.577  -  9.668  =  116.909  meter-kilograms. 

and  the  work  per  second,  since  there  are  45  revolutions  per 
minute,  is 

1^  X  116.909  =  ^  X  116.909  =  87.682  meter-kilograms. 

According  to  experiment,  about  60  per  cent,  of  the  theoretical 
work  is  expended  in  overcoming  friction,  etc.,  so  that  only  40 
per  cent,  remains  effective.     The  effective  delivery  therefore  is 

87.682  X  0.40  =  35.07  meter-kilograms  ]3er  second. 

We  see  then  that  an  Ericsson  engine  which  is  rated  at  one 
horse  power,  gives  in  reality  a  mechanical  effect  of  not  quite 
one-half  of  one  horse  power. 

In  place  of  the  preceding  lengthy  calculations,  we  may  make 

use  of  Equation  XXXIIL,  by  using  the  expansion  ratio  — ^  = 

1.28,  and  multiplying  the  result  by  the  weight  of  air  acting  per 
second.  In  that  equation,  t^  and  ^2  are  the  highest  and  lowest 
temperatures,  and  we  have 

L  =  2.3026  i?  6^  (fi  -  h)  log  1.28. 
L  =  67.4017  G  {h  -  t,)  log  1.28. 

Since  log  1.28  =  0.1072,  we  have 

L  =  7.2255  (^1  -  U)  G. 

If  the  temperature  t^  of  the  outside  air  is  lO'',  and  the  highest 
temperature  is  #1  =  300°,  we  have  ^i  —  ^  =  290^.  Since  now  the 
air  drawn  in  per  second  is  (x  =  0.0424  kilograms,  we  have 

L  =  7.2255  X  290  x  0.0424  =  88.84  meter-kilograms; 
a  value  very  closely  agreeing  with  the  above. 


OPEN  HOT-AIR  ENGINES  WITH  INCLOSED  FIREPLACE.    223 


II.   OPEN  HOT-AIR  ENGINES   WITH  INCLOSED  FIREPLACE.— THE 
HOT  AIR,  AS  BEFORE,  IS  EXPELLED  INTO  THE  AIR. 

In  tlie  machine  already  described,  air  is  necessary  for  com- 
bustion of  tlie  fuel  as  well  as  for  the  action 'of  the  engine.  The 
air  of  combustion  serves  only  to  heat  the  air  in  the  cylinder, 
"without  coming  into  direct  contact  with  it.  The  fire  is  "ea;- 
terior.''  A  large  part  of  the  heat  of  combustion  is  thus  lost. 
As  the  hot  air  is  expelled  into  the  air  after  j)erforming  its  work, 
the  engine  is  "open."  It  is  then  an  "open"  engine  with  "ex- 
terior" fire. 

To  avoid  loss  of  heat,  the  air  of  combustion  and  the  gases  of 
combustion  may  be  used  directly  in  the  working  cjdinder,  in- 
stead of  only  giving  up  a  portion  of  their  heat  to  the  working 
air.  We  have  then  "interior^''  fire,  and  as  the  gases  are  expelled 
after  working,  the  engine  is  still  "ojjen." 

In  the  "open"  engine  then,  with  interior  fire,  the  air  is  first 
compressed  to  a  certain  degree.  It  then  passes  into  an  in- 
closed fireplace  where  its  tension  is  of  course  still  further 
increased,  and  then,  together  with  the  products  of  combustion, 
it  passes  into  the  working  cylinder  and  acts  directly  upon  the 
working  piston.  Such  an  engine  must  evidently  be  an  "open" 
one,  in  order  to  allow  the  products  of  combustion  to  escape. 

When  the  air  comes  in  contact  with  the  fuel,  a  part  of  its 
oxygen  unites  with  the  elements  of  the  fuel  to  form  carbonic 
acid  gas,  carbonic  oxide  gas  and  water.  These  products  of 
combustion  together  with  the  heated  air  remaining  uncom- 
bined,  pass  into  the  working  cylinder.  It  is  evident  that  the 
heat  of  the  fuel  is  thus  much  better  utilized.  The  gases  of 
combustion  have  little  or  no  injurious  effect  upon  the  piston 
or  cylinder  walls,  and  as  the  air  enters  the  fire-box  at  a  greater 
pressure  than  that  of  the  atmosphere,  the  combustion  is  very 
rapid.  It  is  difficult  however  to  prevent  particles  of  coal,  soot, 
and  ashes  from  being  carried  into  the  working  cylinder  and 
injuring  the  working  parts. 

A  description  of  a  large  engine  of  this  kind  will  be  found  in 
Dingier  s  Folytechnisches  Journal,  Band  CLXXXY.,  Heft  6.  It 
was  designed  by  the  engineer  Mazeline,  and  runs  a  paper-mill. 
The  principal  parts  are  represented  in  Fig.  37.     I)  is  the  work- 


224 


THERMOD  TNAMICS. 


ing  cylinder,  with  tlie  piston  D^  and  piston  rod  P.  Tliis  cyl- 
inder is  surrounded  by  another  of  somewhat  greater  diame- 
ter, and  the  cold  air  com- 
ing from  the  feed  cylin- 
der F  passes  through  the 
annular  space  between, 
and  is  thus  warmed  while 
keeping  the  working  cyl- 
inder D  tolerably  cool. 
The  feed  cylinder  has  the 
piston  i^i  and  two  pair  of 
valves  a,  a,  and  h,  h,  of 
which  the  first  allow  air 
to  be  drawn  in  at  every 
stroke,  and  the  others  ad- 
mit the  compressed  air  to 
the  jacket  of  the  working 
cylinder.  After  the  air 
is  here  warmed  and  has 
thus  abstracted  heat  from 
the  working  cylinder,  it 
passes  through  the  pipe 
AzA-i  indicated  by  dotted 
lines,  into  an  inclosed  fire- 
place AA,  which  is  sim- 
ply indicated  in  Fig.  37. 

This  consists  of  an  iron 
cylinder  inclosing  another 
of  fire-brick,  in  which  is  the  grate  and  fire  space.  Above  the 
grate  is  a  funnel  closed  above,  air  tight,  by  an  iron  cover.  In 
this  is  a  cock  which  when  turned  by  special  mechanism,  allows 
the  fuel  to  be  uniformly  spread  over  the  grate.  There  is  also 
an  arrangement  for  stirring  and  shaking  the  grate. 

After  the  air  is  heated  by  the  fire,  and  its  tension  thus  greatly 
increased,  it  passes,  together  with  the  products  of  combustion, 
through  the  space  between  the  fire-brick  and  the  iron  cylinder, 
through  pipe  A-^,  to  the  working  cjdinder  D,  where  it  is  ad- 
mitted, as  shown  in  Fig.  37,  above  and  below  the  piston  by  the 
slide  8S,  precisely  as  in  the  case  of  a  steam  engine.  In  the 
actual  engine,  valves  are  used. 


OPEI{  HOT-AIR  ENamES  WITH  INCLOSED  FIREPLACE.     225 

The  piston  rod  P  is  attaclied  to  tlie  connecting  rod  Q  wliich 
works  the  crank  of  tlie  main  shaft.  This  latter  works  the  con- 
necting rod  and  piston  of  the  feed  or  air  pump  F.  H,  H,  .  .  . 
are  supporting  pillars  of  the  frame,  which  carry  the  cylinders 
and  the  guides  for  the  cross-heads  of  the  connecting  rods. 
Upon  the  main  shaft  we  have  also  the  fly-wheel  V. 

The  working  cylinder  has  a  diameter  of  1.40  meters  and  a 
stroke  of  1.50  meters.  The  diameter  of  the  feed  cylinder  is  1 
meter,  and  the  stroke  the  same  as  that  of  the  working  j^is- 
ton,  or  1.50  meters.  The  cross-section  of  the  latter  is  there- 
fore 

7t  X  1.40  X  1.40      -,  ^oQP 
J =  1.5386  sq.  m. 

and  its  volume 

1.5386  X  1.50  =  2.309  cub.  m. 

while  the  cross-section  of  the  feed  cylinder  is 

TT  X  1.00  X  1.00        .  ^^^, 
J =  0.7854  sq.  m. 

and  its  volume 

0.7854  X  1.50  =  1.178  cub.  m. 

The  volume  of  the  latter  is  therefore  only  little  more  than 
half  of  the  former. 

The  pistons  and  piston  rods  are  kept  lubricated  by  a  special 
arrangement  with  soap-water,  as  oil  or  other  fatty  matter  would 
be  decomposed  by  the  high  temperature  of  the  air  in  the  work- 
ing cylinder. 

In  order  to  set  the  engine  in  action,  a  special  reservoir  of 
compressed  air  is  required,  from  which  air  is  admitted  to  the 
feed  cylinder.  In  the  engine  described,  a  turbine,  which  is  also 
used  in  the  same  mill,  is  made  use  of  for  starting. 

The  action  of  the  engine  is  now  as  follows  : 

The  air  drawn  into  the  feed  cylinder  by  the  first  stroke  is  by 
the  next  stroke  at  first  compressed.  This  compression  is  of 
course  adiabatic,  since  no  heat  is  imparted  or  abstracted  dur- 
ing the  compression.  After  compression  to  a  certain  point, 
and  after  the  air  has  thus  been  heated  by  the  compression  to  a 
15 


226  THERMODYNAMICS. 

certain  temperature,  the  proper  valve  opens  and  allows  tlie 
compressed  air  to  pass  round  the  working  cylinder  to  the  fire- 
place. From  this  point  the  pressure  in  the  feed  pump  is  con- 
stant, and  the  valve  remains  open  until  all  the  compressed  air 
has  been  forced  into  the  fireplace.  As  soon  as  the  first  air 
particles  enter  the  fireplace  they  occupy  a  larger  space,  and 
hence  the  working  piston  begins  its  stroke.  Neglecting  the 
resistances  in  the  conducting  pipes,  this  is  moved  by  the  same 
pressure  as  that  in  the  feed  cylinder  itself.  The  working  pis- 
ton then  begins  its  stroke  when  the  compression  in  the  feed 
cylinder  attains  its  maximum,  or  what  is  the  same  thing,  when 
the  valve  is  forced  open.  The  pressure  in  the  working  cylinder 
also  remains  constant  until  the  feed  piston  has  completed  its 
stroke.  From  this  point  on,  the  working  piston  is  driven  by 
the  expansion  of  the  heated  air,  whose  pressure  at  the  end  of 
the  stroke  must  be  about  one  atmosphere.  This  expansion  is 
also  adiabatic.  As  soon  as  the  working  piston  has  completed 
its  stroke,  the  second  stroke  of  the  feed  pump  compresses  the 
previously  sucked-in  air  before  it,  then  the  other  pressure  valve 
opens,  the  air  enters  the  fireplace,  and  so  on. 

In  this  engine,  which  has  been  examined  by  Tresca,  of  the 
Conservatoire  des  Arts  et  Metiers,  the  maximum  pressure  of  the 
air  in  the  feed  cylinder  was  1.94  atmospheres,  and  the  com- 
pression lasted  up  to  about  one  half  (0.515)  of  the  entire  stroke. 
The  maximum  pressure  in  the  working  cylinder,  on  the  other 
hand,  was  only  1.68  atmos23heres,  which  is  to  be  attributed  to 
the  fact  that  the  pipes  conducting  the  air  from  the  feed  pump 
to  the  furnace  were  too  narrow.  At  0.611  of  the  stroke  expan- 
sion began,  and  the  pressure  fell  gradually  to  the  end  of  the 
stroke,  where  it  was  only  1  atmosphere. 

We  see  from  the  preceding,  that  the  maximum  pressure  in 
this  engine  exceeds  that  in  Ericsson's  only  by  a  small  amount, 
although  according  to  the  views  of  the  constructor,  it  should 
be  5  to  7  atmospheres.  This  pressure  was  most  probably  not 
attained,  because  the  resistances  of  the  driven  machines,  re- 
duced to  the  circumference  of  the  crank  circle,  were  not  great 
enough.  Certainly  the  engine  could  have  produced  a  greater 
effect  than  it  did  during  the  experiments. 


CLOSED  EOT-AIB  ENGINE   WITH  EXTERIOR  FIRE.     227 


III.  CLOSED  HOT-AIR  ENGINE  WITH  EXTERIOR  FIRE. 

The  ]iot-air  engine  may  be  also  "closed,"  in  wliicli  case  the 
same  air  acts  over  and  over.  Thus  the  air,  after  acting  upon 
the  working  piston,  instead  of  being  discharged  into  the  atmos- 
phere, is  cooled  down  to  its  original  temperature,  and  then  used 
over.  Such  an  engine  must  necessarily  haA^e  "  exterior "  fire. 
It  is  analogous  to  a  steam  engine  with  surface  condenser,  in 
which  the  condensed  steam  enters  the  boiler  again,  and  is  used 
over.  To  this  class  belong  the  engines  of  Laubereau  and  of 
Lehmann,  which  will  be  discussed  hereafter.  We  may  also  con- 
sider the  Belou  engine  as  of  this  class.  It  is  at  least  converted 
into  such,  when  we  conduct  the  still  warm  air  which  departs  at 
each  stroke  from  the  working  cylinder  through  a  pipe  sur- 
rounded by  water,  where  it  cools  down  to  its  original  tempera- 
ture before  it  enters  the  feed  cylinder  again. 

Open  hot-air  engines  are  analogous  to  non-condensing  steam 
engines.     The  air  is  discharged  and  a  fresh  supply  taken  in. 


CHAPTEE  IX. 

THEORY  OF  THOSE  OPEN  AND  CLOSED  HOT-AIR  ENGINES,  IN  WHICH, 
DURING  EACH  PERIOD,  THE  AIR  GOES  THROUGH  A  SIMPLE  RE- 
VERSIBLE  CYCLE   PROCESS. 

We  shall  now  give  the  general  theory  for  all  hot-air  engines, 
whether  open  or  closed,  provided  only,  that  in  each  period  the 
air  goes  through  a  complete  reversible  cycle  process. 

There  are  closed  engines,  such  as  those  of  Laubereau  and 
Lehmann,  in  which  only  a  part  of  the  air  is  compressed  or 
rarefied,  while  another  part  is  in  another  condition.  To  such 
engines  the  formulae  which  we  are  about  to  deduce  are  not 
applicable.  For  all  others  in  which  the  entire  volume  of  air  is 
either  compressed  or  rarefied,  our  discussion  holds  good. 

Let  the  area  of  the  piston  in  the  feed  cylinder  be  /,  and  H 
the  length  of  stroke.  Then  the  volume  of  air  drawn  in  per 
stroke  is /If. 

If  t  is  the  temperature  of  this  volume,  measured  by  the  Cen- 
tigrade thermometer,  and  T  the  absolute  temperature,  then 
the  weight  is 

where  j)  is  the  tension. 

Let  this  weight  of  air  be  forced  at  every  stroke  into  the  hot 
space,  and  then  with  an  increased  volume,  due  to  the  heating, 
enter  the  working  cylinder  and  drive  the  piston  there  with 
constant  pressure  through  a  certain  distance,  and  then  act 
expansively  for  the  rest  of  the  way.  If  the  engine  is  open  it 
is  then  discharged  with  tolerably  high  temperature  into  the 
air.  If  the  engine  is  closed  heat  is  abstracted,  until  its  tem- 
perature is  the  same  as  its  original  temperature,  at  the  begin- 
ning of  the  cycle  process. 

Let  the  air  in  the  feed  cylinder  be  compressed  until  its  ten- 

228 


EOT-AIB  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      229 

sion  is  Pi,  and  its  temperature  tlierefore  rises  from  T  to  T-^°. 
"We  liave  then,  from  Equations  XXI«  and  XXI6, 


T 

-af-ifr 

r..r{f)'-. 

....    (2). 

If  T,  p  and  2h  are  known,  T^  can  be  found. 

Now  the  work  L,  which  must  be  performed  in  raising  1  kilo- 
gram of  air  by  compression  from  the  temperature  T  to  T^,  is 
by  Equation  XXII6, 

c 


therefore  the  work  necessary  to  raise  G  kilograms  from  T  to 

Ti°  is 

L,=:^(^T,-t)g (3). 

As  soon  as  the  air  in  the  feed  pump  has  reached  the  tension 
Pi,  the  valves  open,  and  the  compressed  air  is  gradually  forced 
into  the  heating  apparatus.  Here  it  receives  the  temperature 
T^,  at  which,  under  tolerably  constant  pressure  jh,  it  moves 
this  working  piston  until  the  entire  amount  of  compressed  air 
has  been  heated,  and  has  passed  into  the  working  cylinder. 
Then,  as  already  described,  the  air  acts  expansively  during  the 
rest  of  the  way,  until  its  pressure  2^1  lias  sunk  to  p. 

The  work  L^,  performed  by  the  feed  pump  in  forcing  the 
compressed  air  into  the  heating  apparatus,  is,  if  H^  is  that  por- 
tion of  its  stroke  during  which  the  air  is  forced  out : 

L,=fH,p,. 

The  weight  of  air  is  then  also  given  by 

^  _  fSrP, 

hence  we  have 

L^  =  ItT,a,    ......     (4). 


230  THERMODYNAMICS. 

Now,  while  the  feed  piston  is  compressing  the  air  and  forcing 
it  out,  the  atmosphere  helps  it,  and  its  work  is 

But  since  G  =    j^rp   >  we  have 

L,  =  RTG (5). 

Hence  we  have  for  the  work  performed  by  the  feed  piston 
per  stroke 

L,  +  L,-U=0{^^b){T,-T)  .    .     (6). 

Since  c,  ^.and  B  are  constant,  we  see  that  the  work  is  greater, 
the  greater  T^,  or  the  more  the  air  is  compressed,  the  greater 
G,  and  the  less  T. 

Now  let  us  determine  the  work  which  the  hot  air  25ei'forms 
in  the  working  cylinder. 

The  working  piston,  acted  upon  by  the  constant  pressure  p^, 
passes  through  the  same  distance  Hi  as  the  feed  piston. 

If  F  is  the  area  of  the  working  piston,  we  have  the  work 
done 

Li=FHi2Ji. 


But  we  also  have 


hence 


RT., 


L.^RnG (7). 

During  the  expansion,  the  pressure  pi  sinks  to  p),  and  the 
temperature  T^,  ^o  T^.  Since  pi,  p>  and  T2  may  be  considered  as 
known,  T^  is  given  by  the  equation 

or 

Accordingly  the  work  during  expansion  is 

L,--=^{T,-n)G. (9). 


HOT-AIR  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      231 

The  OTercoming  of  tlie  atmospheric  pressure  p  through  the 
distance  Hi^ 

L,  =  FHx>. 
But  since 

RT^  ' 
we  have 

L,  =  IiT,G (10). 

The  effective  work  of  the  hot  air  in  the  working  cylinder  is 
therefore 

L,  +  L,-L,=  G(^^  +  It){T,-T,).    .     (11). 

This  is  greater,  the  greater  G  and  T^  and  the  less  T^g.  If  we 
subtract  (6)  from  (11),  we  have  for  the  effective  deliver}^  per 
stroke 

L  =  g(^^-vr){T,-T,-T,  +  T).    .     (12). 

This  very  simple  formula  shows  that  the  delivery  Ti  of  a  hot-air 
engine,  ivhether  open  or  closed,  in  luhich  the  air  maJces  a  reversible 
cycle  process,  depends  npton  the  temperatures  in  the  engine  and  upon 
the  iveight  of  air  G.     This  weight  of  air  is,  from  (1) 


G 


fHp 
RT' 


and  hence  for  the  same  pressure  %)  and  temperature  TJ,  is 
greater,  the  greater  fH,  or  the  contents  of  the  feed  pump.  It 
increases  also  with  the  tension  %)  of  the  air  drawn  in.  If  the 
atmospheric  pressure  were  3,  4,  or  5  times  greater  than  it  really 
is,  then  for  the  same  cylinder  volume  the  delivery  of  the 
machine  would  be  3,  4,  or  5  times  gTeater,  and  im^ersely,  for 
the  same  delivery  the  volume  of  the  feed  pump,  and  hence  of 
the  working  cylinder,  can  be  3,  4,  and  5  times  smaller.  In  the 
closed  engine  we  may  generate  such  an  artificial  pressure.  We 
have  only  to  compress  the  air  in  the  feed  pump  to  the  required 
degree  before  starting,  and  by  the  completion  of  the  cycle  pro- 
cess bring  it  back  to  this  condition.  In  the  open  engine  this  is 
not  possible.  For  this  reason  it  would  seem  that  the  hot-air 
engine  of  the  future  must  be  of  this  kind,  viz.,  closed. 


232 


THEBMOD  TNAMIC8. 


We  sliall  now  illustrate  tlie  foregoing  considerations  and  cal- 
culations graphically. 

Let  Oa,  Fig.  38, 
I., be  the  volume  of 
air  drawn  in  per 
stroke  by  the  feed 
pump,  of  pressure 
p  and  temperature 
T.  Let  this  vol- 
ume be  compress- 
ed adiabatically 
from  h  to  c,  the 
pressure  rising  to 
2^1  and  the  tem- 
perature to  Ti. 
When  the  press- 
ure pi  is  reached, 
the  valve  opens 
and  the  air  is 
forced  out  under 
constant  pressure 
^j*!.  The  work  per- 
formed during  this 
operation  is  evi- 
dently given  by 
the  area  abcdO. 
But  the  outer  air 
has  performed  the 
work  a?^eO.  Hence 
the  shaded  area 
S  5cr?e  gives  the  work 
of  the  feed  pump 
in  compressing  the  air  and  forcing  it  out  into  the  heating 
apparatus. 

In  the  heating  apparatus  the  temperature  rises  to  T.^  and  the 
volume  is  increased.  The  increased  volume  is  given  by  fg  in 
Fig.  38,  n.  As  soon  as  this  volume  is  reached,  the  air  expands 
adiabatically  from  g  to  h  and  its  volume  is  now  01.  During 
expansion  the  pressure  2h  sinks  to  p,  and  the  absolute  temper- 
ature Ti  to  T^.     The  entire  work  performed  by  the  working 


EOT-AIB  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      233 

piston  is  given  by  tlie  area  fgMO.  But  this  piston  lias  had 
to  overcome  the  pressure  of  the  atmosphere  p,  which  requires 
the  work  IhiO.  Hence  the  shaded  area  fghl  gives  the  effect- 
ive work  of  the  working  piston.  If  from  this  we  subtract  the 
work  of  the  feed  piston,  we  obtain  the  effective  work  of  the 
machine,  or  that  work,  part  of  which  goes  to  useful  effect  and 
part  to  overcome  the  prejudicial  resistances.  This  work  is  then 
given  by  the  shaded  area  T^T^T^T  (HI.),  inclosed  by  two  adia- 
batics  and  two  straight  lines.  In  Chapter  YI.  the  cycle  pro- 
cess consisted  of  two  adiabatics  and  two  isothermals.  But 
there,  during  expansion  the  air  sank  to  its  original  temperature, 
while  here  it  sinks  only  to  a  higher  temperature  T^. 

Let  us  now  deduce  formulae  for  the  dimensions  of  the  hot- 
air  engine. 

If  we  substitute  in  (12)  -^4r  iii  place  of  G,  we  have 


Since,  according  to  Equation  YII., 

and  hence 

L  =  3.439  =?P^  (7T,  _  51  -  Ti  +  T). 

If  the  engine  makes  %  strokes  per  minute,  the  delivery  per 
second  is 

L,  =3.439  ?^ .  ^  (7^,-  7^3  _  51  +  T). 
The  delivery  in  horse  powers,  is  then 

^ = m  -?"  (^^  -  ^=  -  ^' + ^)  •  (™i-> 


234  THERM0DTNAMIC8. 

wliere  p  is  tlie  atmosplieric  pressure  in  kilograms  per  square 
meter.     If  2^  is  given  in  atmosjDlieres 


N  =  7.8972  =?^^ {T.- T,-T,+  T)  horse  powers.     (XLIII) 

If  we  consider  N  as  given,  we  liave  for  the  volume  FH  of  tlie 
working  cylinder 

T.N 
^^-^  7.8972pn(T,-T,-T,  +  T}  ^^^^  '^"'«'^^-  '    <^^^^-> 

Hence  we  see  that  the  contents  of  the  working  cylinder  are 
not  only  less  as  the  pressure  p  increases,  but  also  the  greater 
the  number  of  revolutions. 

Let  us  now  determine  the  relation  between  the  area  of  the 
working  and  feed  pistons. 

The  weight  G  of  air  drawn  into  the  feed  cylinder  at  every 
stroke,  of  T°  temperature  and_^  atmospheres'  pressure,  is 

BT  ' 

This  same  weight  of  air  fills,  after  expansion,  the  entire  work- 
ing cylinder.     Hence  we  have 

BTs' 

If  the  two  weights  are  equal,  we  have 


^       ^      or    /  =  i^^^     .     .     .     (XLV.) 


We  have  also 


hence 


Q^fJ^^P^    and     G-™^^' 


f       T,    ""'  -^     -^  T, 


EOT-AIB  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      235 


Since/  is  the  same  in  botli  cases, 
T       71 


T,       T, 


or     T^:T,'.'T^:T, 


T^  X  T=T^x  T^. 


(XLVI.) 


Since  T-i  is  tlie  highest,  and  ^the  lowest  temperature  of  the 
air  during  the  process,  and  ^3  and  2\  are  the  intermediate  tem- 
peratures, we  have  the  important  principle, — ivlien  a  given  quan- 
tity of  air  goes  through  a  cycle  of  the  Moid  in  question,  the  j»^oduct 
of  the  extreme  temperatures  is  equal  to  the  product  of  the  interme- 
diate temperatures. 

In  deducing  this  princijDle,  we  have  assumed  that  the  air  is 
compressed  in  the  feed  cylinder,  and  forced  into  the  heating 
apparatus  under  the  constant  pressure  p^,  and  that  the  heated 
air  drives  the  working  piston  with  the  same  pressure  p],  through 
the  distance  H^.  In  short,  that  the  air  is  heated  under  con- 
stant pressure.  By  the  aid  of  the  higher  mathematics  the  cor- 
rectness of  this  principle  may  be  proved,  for  any  case  whatever, 
where  the  heating  takes  place  according  to  the  general  law, 

if  only  the  rise  and  fall 
of  temperature  are  pro- 
portional to  the  heat  im- 
parted  or  abstracted. 
Let  us  seek  to  show  the 
correctness  of  this  by  a 
practical  example,  in 
such  a  way  that  it  will 
be  evident  that  the  same 
proof  will  hold  good  for 
every  other  special  case. 
Let  Ov  =  V,  Fig.  39, 
be  the  specific  air  vol- 
ume, its  tension  p)  and 
tempei-ature  T.  Suppose  heat  is  imparted,  and  the  air  expands 
while  performing  work  along  the  curve  TcTg,  to  the  condition 
Vzp%T~;,.    Let  the  heat  thus  imparted  be  Q. 


236  THEBMOD  YNAMIGS. 

Let  us  assume  that  the  exponent  —  has  the  value  —  2.     We 

^  m 

further  assume  the  volume  v^  =  ft'. 
"We  can  easily  compute  the  tension  ^h,  since  "Cipi  are  known. 
Suppose  D  —  1,  p  —  1.     Then  by  the  law, 

lxl^  =  p^(-^     or    _p2  =  jg  =  1.5625. 

If  the  absolute  temperature  T  is  273  +  100  =  373^  we  can 
easily  find  Tg.     Thus  by  Equation  XXXY., 


Ta  =  373  (I)   =  373  x  1.9531  =  728.5°. 

If  more  heat  had  been  added,  until,  for  example,  the  volume 
Vi  =  \T)  or  2y,  the  tension  ^3  and  temperature  T<^  would  have 
been  greater. 

The  specific  heat  is 

mk  —  n 

s  = c, 

m  —  n 

or  putting  for  m,  n,  k,  and  c  their  special  values, 

s  ■=  -^f -*-  ~  ^  ~,^^  0.16847  =  0.1916. 

Let  us  pass  through  the  point  T2,  determined  by  Vg  =  f  v  and 
p,  =  1.5625p,  an  adiabatic  curve,  which  we  shall  call  the  adia- 
batic  curve  of  the  point  T^.  We  may  construct  this  curve  from 
the  formula 

p^v^^  =  psVs^  =  piV^,  etc., 

by  substituting  various  values  for  v^,  v^,  etc.,  and  finding  the 
corresponding  pressures.     In  Fig.  39,  this  curve  is  Ta^s^^i- 

Now  let  us  suppose  the  air  of  volume  Vg  and  pressure  ^2  and 
absolute  temperature  T^,  to  expand  adiabatically  until  its 
pressure  falls  to  ps,  which  we  may  suppose,  for  example,  to  be 
=p.     We  can  then  easily  find  the  volume  v^  and  temperature 


HOT-AIR  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      237 

Tq,  wliicli  tlie  air  possesses  after  expansion.     (S^q  might  have 
supposed  V3  given,  and  then  found  JJ3  and  Tg.) 
We  find  2h  from  the  equation 

p^vi  =  2hVs'', 


1.5625  (1-)^-^^  =1  X  ^3'-^^  or  log  1.5625  +  141  log  f  =  1.41  log  V3, 

heace 

-ya^  1.7155. 

As  soon  as  we  know  V3,  we  can  find  T3  from  the  formula 

T.       /  1.25X0.^^      ^^^^3. 


T,       U.7155y' 

During  expansion,  therefore,  the  absolute  temperature  sinks 
from  728.5  to  469.8°. 

Let  now  the  air  of  volume  V3  and  pressure  2h  and  absolute 
temperature  T^  be  compressed  to  the  volume  of  t\,  pressure  2^1, 
and  temperature  T^,  such  that  the  point  Ti  lies  on  the  adia- 
batic  through  T,  and  let  this  com2Dression  take  place  according 
to  the  same  law  as  the  expansion.  It  is  evident  that  a  certain 
quantity  of  heat  must  be  abstracted,  in  precisely  the  same 
manner  as  before  heat  was  imparted.  That  there  is  abstraction 
of  heat  and  expenditure  of  work,  is  seen  from  the  curve  T-^aTx 
approaching  the  axis  from  above  from  right  to  left. 

Let  us  determine  now  the  pressure  pi,  the  volume  v^,  and  the 
temperature  T^  of  the  air  when  it  reaches  the  adiabatic  ToT^. 

We  have  for  the  curve  T^aT^ 


and  for  ToT^ 

From  (2)  we  have 


F3V3~'=i3lVl~^     ......        (1). 

^^k  —  p^yk    .......       (2). 


p^=p{l) 


and  putting  this  value  in  (1) 


iV^'3      '=i^  ^73:41       or       ^1         =  ,, 


238  THEBMODYNAinCB. 

If  in  our  special  case,  ^jg  =jp  =  1  and  v  =  1  and  Vg  =  1.7155 

v^^'^  =  (1.7155)2  =  2.743     or    v,  =  1.3725. 
Since  now  we  know  v^  and  i\  we  can  find  T^.     Tlius 

T,  _  fVsY'^  /1.7155\~' 
T3  ~  \vj      ~  V1.3725/ 
or 

Ti  =  240.2°. 

Tliiis  tlie  absolute  temperatures  are 

373,     728.5,     469.8,     240.2. 


Hence 


T,      728.5      T  Q^         ,      T,      469.8      ,  „. 


This  is  the  same  result  which  we  have  already  obtained  for 
heat  addition  and  subtraction  according  to  a  straight  line  or 
according  to  the  law 


The  heat  Q  imparted  on  the  path  TcT^  is 

0.1916  (728.5  -  373), 
or  generally 

s{T.^-  T)  heat  units. 

The  heat  abstracted  on  the  path  T^aTi,  is 

Q,  =  0.1916  (469.8  -  240.2), 

or  generally 

s(T3-  Tt)  heat  units. 

Since  now  from  the  proportion 

T,'.T::T,'.  T„ 
we  have 


HOT-AIR  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      239 
we  at  once  obtain 

tliat  is 

Which  is  the  same  principle  proved  in  Chapter  V.,  with  iso- 
thermal lines  and  adiabatics.  We  should  obtain  the  same  re- 
sult if  we  take  —  equal  to  —  3,  —  4,  —  5,  or  generally  equal  to 

any  number,  positive  or  negative,  whole  or  fractional. 

It  is  now  easy  to  determine  the  points  T  or  T^  in  which  the 
isothermal  lines  TdT  and   T^gT^  intersect  the  adiabatic  line 

To  determine  the  point  T  we  have 

23V  =  qiv (1). 

and 

p^V2^  =  qw^' (2). 


From  (1)  we  find  q  =-^— ,  and  inserting  this  in  (2) 


JJV 

id'-' 


pv 
Thus  in  the  case  of  our  special  example, 

^yo-"^  =  1.5625  (!)'■''  or     lu  =  6.384, 
V,  of  course,  being  1.     We  have  also 

when  j3  =  1. 

In  like  manner  for  iv^  we  have 

0  41       PsVs^'^^  -,  Pll^l 

-ic;  "-41  =  ^elJ —     and     Qi  =  -^-^—  . 

PlV,  u\ 

If,  then,  in  any  closed  or  open  hot-air  engine,  in  which  the 
air  makes  a  reversible  cycle,  the  air  compressed  in  the  feed 
cylinder  is  not  heated  under  constant  pressure  ;  whatever  may 
be  the  law  of  heating — whether  under  constant  volume,  as  in 
Sterling's  engine,  or  according  to  any  law,  as^i^r^  =pv~^,  as 


24:0  THEBM0DTNAMIG8. 

shown  by  the  curve  7c 7^2  in  tlie  Figure — the  ]7roduct  of  the  tioo 
extreme  temperatures  is  ahvays  equal  to  the  product  of  the  mean,  if 
heat  is' abstracted  according  to  the  same  law  as  it  is  imparted. 

The  formulse  which  we  have  developed  thus  far,  therefore, 
apply  to  all  kinds  of  open  and  closed  hot-air  engines,  provided 
that  in  the  latter  the  expansion  is  such  that  by  the  subsequent 
compression  and  cooling,  the  air  returns  to  its  original  condi- 
tion ;  and  provided  that  in  the  former  the  air  escapes  into  the 
atmosphere  under  the  assumed  conditions. 

Transformation  of  our  Formulce. — Since  in  the  formula  al- 
ready deduced 

we  may  put  ^ :,  in  place  of  — r^ ,  the  equation  can  be  written 

-       ^^^'  -^[_ch{T,~T,)-ch{T,-T)-\. 


c{k-l)  2\ 


Now  ch(T2—  T-^  is  the  amount  of  heat  added,  and  clc{T^—  T) 
that  abstracted — in  case  of  an  open  engine,  that  which  is  given 
up  to  the  air.  Also  ck  is  the  specific  heat  for  constant  pressure, 
or  for  the  law 

pv^  —  p^Vi  —  etc. 

But,  as  we  have  seen,  our  formulae  also  apply  when  the  change 
of  volume  and  pressure  is  given  generally  by 

If  s  is  the  specific  heat  generally,  we  have  then 

whereas  we  know 

mk  —  n 


EOT-AIB  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      241 

If  tills  engine  makes  n  strokes  per  minute,  we  have  tlie  horse- 
power per  second 

^=  60^  •  c{k-l)T,  ^^'  -  ^^  -  ^3  +  ^)' 

where  p  is  the  pressure  in  kilograms  per  square  meter. 
Jlp  is  given  in  atmospheres,  we  have 

^j_    10334??  FHpS  .rp  rp  rp     ,      rp. 

or 

N=  2.296  ^^^l^rj.  (^2-  T,-T,+  T)  horse-power.  (LI.) 


Inversely,  the  volume  FH  of  the  working  cylinder  is 
2.296pis  {T^-Ti-l\-\-  T) 


FH^  ._     ±-^^'1       ^,  cubic  meters. 


or  since 

c  (7^-1)  =  0.06904, 

^^  0.03007^^3  ..         ,  ,-^TT^ 

Maximum  Delivery  of  the  Hot-Air  Engine. — If  it  were  possi- 
ble in  practice  to  raise  at  will  the  temperature  of  the  air  in  a 
hot-air  engine  as  high  as  we  please,  we  might  construct  such 
engines  with  smaller  dimensions  and  relatively  higher  delivery, 
and  then  probably,  since  they  are  not  liable  to  explosion,  they 
might  replace  the  steam  engine.  But  since  the  temperature  is 
practically  limited,  this  is  not  the  case. 

We  may  consider  the  extreme  temperature  to  which  the  air 
may  be  raised,  as  about  300"  C,  although  this  has  been  occa- 
sionally exceeded  by  as  much  as  30°  without  injury. 

If  we  assume,  then,  300"  as  the  maximum  temperature,  and 
further  assume  that  the  temperature  of  the  air  on  entrance,  or 
in  case  of  a  closed  engine,  at  the  beginning  of  a  cycle,  is  0°, 
then  in  our  formulge  we  have  T^  =  273  +  300  =  573  and  T  = 
273  +  0  =  273.  Now  for  the  maximum  delivery,  or  the  least 
volume  of  cylinder  for  given  delivery,  the  intermediate  tem- 
16 


242  THERMODYNAMICS. 

peratures  T^  and  Tz  must  liave  definite  values.  Thus,  it  is 
clear  that  the  compression  in  the  feed-pump  cannot  be  carried 
so  far  as  to  raise  the  temperature  to  300"  C,  because  then  no 
heat  could  be  imparted  by  the  heating  apparatus,  and  there- 
fore no  mechanical  work  could  be  obtained.  Thus,  if  we  had 
Ti  =  Ta,  we  would  also  have,  according  to  L.,  T  =  T^,  and  the 
expression 

would  be  zero.  For  the  same  reason  we  cannot  have  T^  =  T, 
for  then  from  L.,  T^—  T^,  and  the  above  sum  is  zero.  The 
value  of  Ti  must  then  lie  somewhere  between  T  and  T2.  Now 
the  expression  T^  —  T^—  Ti+  T  will  be  a  maximum,  when 
^1  +  J's  is  a  minimum.  Also,  the  product  of  T^  and  T^  is  con- 
stant and  equal  to  573  x  273,  or  T^  x  T.  We  can  easily  find 
then,  by  calculus,  that  T^  +  T'3  is  a  minimum  when  1\  =  T^. 

If  now  Ti  =  T^,  we  have  from  L., 

T^T  =  T,^  =  Ti,  hence  T^  =  T,  =  VT^=^  V573x273 
=  395.51.  The  intermediate  temperatures  Centigrade  must 
then  be,  h  =  h  =  395.51  -  273  ^  122.51°.  If  we  insert  these 
values  in  LII.,  we  have  for  the  volume  of  cylinder  FH  when 
the  delivery  is  a  maximum, 


Inversely 


^^^02163^ 

piis  ^  ^ 


^^FHpns 

0.2163  ^         ' 


If  the  heat  is  added  or  abstracted  under  constant  pressure, 
then  s  =  0.2375  and 


hence 


i^jg^=  0.9107  — (LV. 

P7l 


i\r=  ^^ (LVI. 


0.9107 
If,  as  in  Sterling's  engine,  heat  is  imparted  or  abstracted 


HOT-AIR  ENOINE  WITH  COMPLETE  CYCLE  PROCESS.      243 

under  constant  volume,  s  =  0.1685.  In  this  case,  the  volume 
at  the  end  of  expansion  is  equal  to  that  at  the  beginning  of  the 
cycle  process.  The  air 
of  volume  Fand  press- 
ure p,  as  in  Fig.  40, 
is  first  compressed 
adiabatically  to  Fj  and 
'Pi.  Then  the  volume 
Fj  is  heated  from  Ti  to 
Ta,  and  its  pressure 
rises  from  p^  to  2h- 
The  volume  V^  now  ex- 
pands adiabatically  to 
its  original  volume  V, 
the  temperature  fall- 
mg  from  l\  to  7 3,  and 

pressure  from  p^i  to  p^,  or  for  the  maximum  delivery  T^  falls  to 
J'l.  Finally  heat  is  abstracted  from  v  till  T^  or  T^  falls  to  T, 
and  px  to  p. 

If  the  heating  and  cooling  takes  place  according  to  the  law 


we  have 


s  =  0.1916. 


If  in  a  Sterling  engine  the  original  volume  is/ZT,  we  have 


G  =  '^Tp  )  and  in  LII.  we  have  T  in  place  of  T^. 


Hence 


Inversely 


0.03007  X  273  X  i\r  _ 
-^        pnx  0.1685  X  54.98" 

fHpn 


pm 


N-- 


0.886 


.     (LYII) 


(LVIII. 


We  see  that  the  delivery  in  all  cases  is  greater,  and  hence 
the  final  volume  of  the  air  at  the  end  of  the  cycle  is  less,  the 
greater  the  initial  pressure  p  and  the  number  of  strokes  n  per 
minute. 


244  THERMODYNAMICS. 

Heat  Imparted  and  Abstracted — Consumption  of  Fuel. — If  in 
formula  L6,  viz., 

we  put  AR  m  place  oic{k  —  1),  we  have 


FHp 


a 


i  =  ^  [s (^2  -  T,)  -  s  (^3  -  r)  ]  meter-kilograms. 

Here  Gs  [T^  —  T^  is  tlie  lieat  added  to  the  air  per  stroke, 
and  6^5  (Tg  —T)  is  the  heat  abstracted, or,  if  the  engine  is  open, 
the  heat  given  up  to  the  outside  air.  If  we  denote  the  first  by 
Q  and  the  second  by  ^i  we  have 

Q^s{T^-T^G (LIX.) 

Q,^s{T,-T)G (LX.) 

If  there  are  n  strokes  per  minute,  or  60?i  per  hour,  the  heat 
imparted  per  hour  is 

Qn  =  60?is  (^2  -  71)  G  heat  units  .     .     (LXI.) 

and  that  abstracted  is 

q,n  =  ^^ns{T^-T)G.    .     .     .     (LXII.) 

For  the  maximum  delivery,  T^  -  T^  =  573  -  395.51  =  177.49 
and  T,-T=  395.51  -  273  =  122.51,  hence 

Q;,  =  1064:9 AnsG  =  9509  ns23FII  heat  units. 

Q,,^=    7350.6nsG=-6563nspFH     " 

where  p  is  given  in  atmospheres. 

We  have  therefore  jxr  liorse-poioer  per  hour 

Q^-^509nspFH 

-^>-  —  ^^= iicdit  uiiiuo. 

N  M 

%=556?!S^  teat  units. 


HOT-AIR  ENGINE  WITH  COMPLETE  CYCLE  PROCESS,      245 

If  one  kilogram  of  coal  furnislies  7,500  heat  units,  tlie  com- 
sumption  of  fuel  per  liour  for  every  horse-power  is 

kiloarams, 


if  all  the  heat  is  utilized.     If,  however,  half  is  lost  by  radiation, 
the  consumption  per  horse-power  per  hour  is 

Qn 


3750^ 


kilograms    ....     (LXIII.) 


Example  I. — What  is  the  delivery  of  a  hot-air  engine  in  which  FH  =  2.309 
cubic  meters,  n  =  46,  p^  =  1.94  atmosplieres,  and  the  worJdng  pressure ^2  =  1-68 
atmospheres,  when  T=27S  +  10  =  283"  ? 

According  to  XXIa  and  XXI&, 

/1.94\o-59<" 
Ti  ==  283  f  -J-  J         =  343°. 

From  experiments  upon  this  engine,  the  escaping  air  was  found  to  have  a 
temperature  ^3  =  250°,  or  an  absolute  temperature  T3  ■—  273  +  250  =  523' . 
Hence  the  temperature  To  in  the  working  cylinder  is 

T2  =  523  (l.C8)''-290'  =  608.25°. 
Substitute  in  XLIII.,  and  we  have 

N=  7.8972  ^'^^^.Q  ^^  (608.25  -  523  -  343  +  283) 

=  40.5  horse-power. 

The  horse-power,  as  actually  found  by  Tresca  by  means  of  the  indicator,  was 
about  40. 

If  we  assume  that  30  per  cent,  of  this  theoretical  delivery  is  consumed  by  fric- 
tion, we  should  have  0.70  x  40.5  =  28.35  effective  horse-power.  The  work  of  a 
steam  engine  of  the  same  dimensions  with  3  or  4  atmospheres'  pressure  would  be 
much  greater.  But  if  we  take  in  the  boiler  and  all,  the  space  occupied  is  in  favor 
of  the  hot-air  engine. 

The  weight  of  air  G  used  per  stroke  is 

EHp     2.309  X  10334      23861      ,  ^^^  ,  ., 

R^  =  29.272  X  523  =  15309  =""  ^'^^^  ^^^^S^'""^^' 

The  heat  imparted  per  stroke  is  therefore  by  LIX. 

Q  =  0.23751  (608.25  -  343)  1.558  =  98.149  heat  units, 

and  the  heat  imparted  per  horse-power  per  hour  is 

98.149  X  46  X  60      270894      „„„„  ,      ,       ., 
—;;:.—-  =  6689  hoat  units. 


40.5  40.5 


246  THERMODYNAMICS. 

Hence  the  expenditure  of  coal  per  hour  per  horse-power,  if  all  the  heat  were 
given  up  to  the  air,  would  be  ff^-^  =:  0.892  kilograms.  According  to  Tresea's 
experiments,  the  consumption  was  1.44  kilograms,  and  therefore  about  ^-  of  the 
heat  was  lost. 

The  best  steam  engines  use  as  low  as  1,  and  at  most  3  kilograms  of  coal  per 
hour  per  effective  horse-power.  The  consumption  for  the  hot-air  engine  is  thus 
somewhat  greater  than  for  the  best  steam  engines. 

Example  2. — Required  to  construct  a  hot-air  engine,  so  that  the  cylinder 
volume,  for  a  given  delivery,  shall  be  a  minimum.  What  should  be  the  volume 
if  the  theoretical  delivery,  for  48  strokes  per  minute,  is  to  be  100  horse-power  ? 

According  to  LV.,  we  have 

FH  =  0.9107  -^^  =  ^^  =  1.9  cubic  meters. 
1  X  4o  4o 

A  steam  engine  working  without  expansion  or  condensation,  under  3^  atmos- 
pheres, with  48  strokes  per  minute,  would  require  only  one  cylinder  of  0.368 
cubic  meters  contents. 

If  the  cylinder  of  our  hot-air  engine  is  not  to  be  greater,  we  must  have  a 
closed  engine,  and  make  the  initial  pressure  of  the  air  somewhat  more  than  5 
atmospheres.  The  compression  in  the  feed  cylinder  would  then  raise  the  press- 
ure to  about  16  atmospheres.  These  are  pressures  which  certainly  can  hardly 
be  recommended  in  practice.  A  cylinder  whose  volume  is  twice  or  three  times 
0.368  cubic  meters,  can  hardly  be  called  excessively  large,  however.  The  cylin- 
der of  the  early  condensation  engines  of  Watt  had  for  an  effective  pressure  of 
I'j  atmospheres,  and  a  theoretical  delivery  of  120  horse-power,  a  volume  of  1 
cubic  meter.  To  this  was  added  the  immense  boiler,  which  is  wanting  in  the 
hot-air  engine. 

If  the  air  in  our  machine  is  compressed '  before  admission  to  2  atmospheres, 
the  working  cylinder  for  the  same  delivery  will  need  to  be  only  half  the  size,  or 
about  0.95  cubic  meters,  that  is,  about  3  x  0.368  cubic  meters,  and  the  pressures 
in  the  engine  will  not  be  excessive. 

From  all  this,  it  appears  that  even  such  hot-air  engines  as  have  a  theoretical 
delivery  of  100  horse-power,  and  therefore  an  effective  power  of  50  to  60  horse- 
power, are  not  without  a  probable  future,  only  they  must  be  closed  engines,  ivhose 
initial  pressure  is  2  or  3  atmospheres,  a?id  they  must  be  constructed  for  the  maxi- 
mum delivery.  Especially  to  be  recommended  are  smaller  engines  (8  to  20  horse- 
power), because  they  make  in  the  same  time  more  revolutions  per  minute, 
and  occupy  a  relatively  less  space  than  the  larger.  It  is  worth  noticing  that 
Zeuner,  in  his  "AVarme  Theorie,"  has  found  for  the  cylinder  volume  results 
double  of  those  here  given.  The  reason  is,  that  he  considers  only  single-acting 
engines.  If  in  his  formulas  we  substitute  2u  in  place  of  u,  which  denotes  the 
number  of  revolutions  per  minute,  the  results  coincide. 

Let  us  now  comiiute  for  our  engine  above,  of  100  horse-power,  the  volume  of 
the  feed  cylinder,  as  well  the  consumption  of  fuel,  and  the  amount  of  cooling 
water,  assuming  that  the  machine  is  closed. 

We  have  already  found 

fll=  FII^  =  1.9  ^-^^  =  1.9  X  0.6902  ='1.311 
cubic  meters  =  the  vohime  of  the  feed-pump. 


HOT-AIR  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      247 

The  amount  of  heat  added  per  hour  is 

Qh=  9509)ispFH, 
or  inserting  numerical  values 

Q„  =  9509  X  48  X  0.2375  x  1.9  =  205851  heat  imits. 

Hence  the  consumption  of  coal  pev  hour  for  each  horse-power  is 

Qk  205851 


3750i\r      3750  x  100 


0.549  kilograms. 


Since  the  effective  delivery  is  perhaps  at  most  60  horse-power,  each  effective 
horse-power  per  hour  would  require 

0.549x100      „„,^,., 

j^ =  0.915  Kilograms, 

or  a  less  quantity  than  the  best  steam  engines. 

The  heat  abstracted  per  hour  is  found  from  the  proportion 

^  =  9509  :  6563,  or  since  a.  =  205851 

20581  :  ^1 A  =  9509  :  6563,     or 

g,ft  =  142037  heat  units. 

If  the  cooling  water  is  heated  from  0  to  80°,  we  must  have  per  second 

142036 
3-60^^80  =^-^^^^^^^°Srams, 

1  08 
or  in  English  weights,  about  1.08  lbs.,  or  ^^^  cubic  feet.     That  heat  is  here  not 

taken  into  account  which  is  abstracted  from  the  air  by  partial  evaporation  of 
the  water. 

The  Regenerator. — Let  us  now  consider  an  apparatus  applied 
by  Ericsson  in  liis  first  caloric  engine,  which  he  calls  the  "  re- 
generator." 

In  the  engine  already  discussed,  we  saw  that  the  air  esca23ed 
with  a  temperature  of  260°.  It  escaped  with  almost  the  same 
temperature  from  Ericsson's .  first  engine.  It  was  evidently 
desirable  to  utilize,  at  least  partially,  this  heat,  and  thus  econ- 
omize fueL  He  therefore  caused  the  escaping  air  to  pass 
through  a  network  of  wires,  which  thus  became  heated.  The 
cold  air  drawn  in  by  the  feed-pump  then  passed  through  the 
apparatus,  and  was  thus  heated. 

Now  from  formula  12,  page  23,  we  have  seen  that  the  de- 
livery depends  especially  upon  the  weight  of  air  G.  This 
weight  depends,  however,  whether  the  air  is  compressed  or 
not,  upon  the  temperature.     The  greater  the  temperature,  the 


248  THERMODYNAMICS. 

less,  for  equal  volume,  tlie  weiglit.  Therefore,  under  similar  cir- 
cumstances, the  working  cylinder  receives  a  less  weight  of  air 
and  performs  less  work.  For  the  same  delivery  the  volume  of 
the  working  cylinder  must  be  increased.  For  this  reason  the 
use  of  the  regenerator  gave,  in  Ericsson's  first  engine,  a  less 
delivery  than  without,  and  this  may  be  the  reason  of  its  omis- 
sion in  his  later  engine. 

If,  however,  the  value  of  a  hot-air  engine  is  estimated,  as  is 
proper,  by  the  ratio  of  fuel  consumption  to  the  dimensions  and 
delivery,  the  use  of  the  regenerator  is  advantageous  ichen  the 
engine  does  not  give  the  maximum  delivery.  If,  for  example,  the 
expansion  is  not  carried  so  far  that  the  temperature  sinks  to 
Ts  =  122.51°  C,  but  only  say  to  T3  =  160°  C,  the  heat  160  - 
122.51  =  37.49°  can  be  added  by  the  regenerator.  The  deliv- 
ery will  be  less  than  the  maximum,  but  in  the  same  degree  less 
fuel  will  be  needed.  If  this  37.49'  is  lost,  not  only  will  the 
delivery  be  less  than  the  maximum,  but  also  more  fuel  will  be 
necessary.  It  follows,  then,  that  in  a  hot-air  engine  ivhich  gives 
the  maximum  delivery,  ivhether  open  or  closed,  the  regenerator  is  of 
no  effect ;  but  in  one  which  is  not  so  arranged  as  to  give  the 
maximum  delivery,  it  may  be  advantageous. 

Absolute  Maximum  Delivery. — The  absolute  maximum  deliv- 
ery of  a  hot-air  engine  can  only  be  attained,  when  the  cycle 
process  is  of  the  character  described  in  Chaj^ter  VI.,  in  which 
case  the  heat  imparted  for  a  certain  work  is  least.  Therefore 
the  addition  and  abstraction  of  heat  must  be  so  regulated 
that  the  air  expands  in  the  working  cylinder,  at  first,  according 
to  the  isothermal,  and  then  according  to  the  adiabatic  curve, 
and  that  the  compression  in  the  feed-pump  should  also  be  simi- 
lar.    The  law  of  addition  and  abstraction  of  heat  is  therefore 

pv  =  2^iVi . 

If  we  wish  a  formula  for  the  delivery  of  such  a  machine,  we 
must  put  Ti  =  T2  and  ^3  =  T  in  the  equation 

L  =  ^ls{T,-T,)-s{T,-T)]     on  page  244, 

since  the  temperature  or  inner  work  is  constant  during  the  re- 
ception and  abstraction  of  heat.     If,  however,  T^  =  T2  and  T^  = 


HOT-AIR  ENGWE  WITH  COMPLETE  CYCLE  PROCESS.      249 

T,  we  cannot  find  the  delivery  from  tlie  equation.  We  know, 
however,  that  under  the  given  circumstances  we  can  deter- 
mine the  heat  imparted  and  abstracted  from  the  initial  and 
final  volumes,  or  from  the  initial  and  final  temperatures.  For 
the  heat  imparted  we  have,  page  158, 

Q  =  2.3026^i?ri  log  ^,   if  v^  is  the  initial  volume  at  the 

greater  pressure  2h ,  and  v^  the  final  volume  after  isothermal  ex- 
pansion. 

For  the  heat  abstracted 

^1  =  2.3026  ^i^Ta  log—', 

where  v^  is  the  greater,  and  V4  the,  less  volume. 

We  can  now  find  the  work  L  according  to  Equation  XXX., 
page  184.     We  have 

for  one  kilogram  of  air.     For  G  kilograms  we  have 

Putting  for  Q  its  value  we  obtain 

L  =  2.3026EG{Z  -  T,) log  — ^ 

This  is,  then,  the  absolute  maximum  delivery  of  a  hot-air  en- 
gine.    If  we  denote  it  by  La,  we  have 

La  =  2.3026EG  {T,  -  T,)  log  — ^ 

If  we  express  the  weight  G  in  terms  of  the  cylinder  volume, 
we  have,  page  228, 

where  ih  is  the  final  pressure  and  T^  the  final  temperature  after 
expansion  in  the  working  cylinder.     We  have  therefore 

La  -  2.d026FHps  ^^-^'  log  -'. 


250  THERMODYNAMICS. 

But  we  have  —  =  — ,  and  accordina;  to  the  law  of  the  adia- 
batic  curve 

hence 

k 


If  we  substitute  this  value  of  ^2  in  the  expression  for  —,  we 
have 

V,  'ps\rj    ' 

Hence 

T  —   T  11    /  T  \  3.439 

La  =  2.3026i^^3       rr       log  —(4-')       meter-kilograms. 
If  the  strokes  per  minute  are  n,  the  delivery  per  second  is 

V)  T   —   T  n     /  7^X3.439 

^  X  2.3026i^i?p3  ^^-^  log  ^  (  41 )       meter-kilograms, 
and  the  horse-power  is 

If  the  pressure  is  given  in  atmospheres,  we  have 
10334  X  2.3026 


X  75 


iFHpz,  etc.,    or 


N  =  5.288nFH2J3  ^S„  ^'  log B  ( ^^ )       horse-power. 
-^2  Pz  ^  J-v 

If  we  take  as  outer  limits,  Ti  =  573  and  T^  =  273,  we  have 
i^^=581.15i^^^3?^log       ^^' 


12.804233 

If  this  value  is  real,  we  must  have 
Pi>  12.804^93, 
that  is,  the  pressure  for  the  least  volume  Vj  must  be  at  least 


EOT-AIR  ENGINE  WITH  COMPLETE  CYCLE  PROCESS.      251 

greater  tlian  12.804  times  tlie  initial  pressure  p^.     If,  then,  we 
take  pi  =  20  and  ps  =  1  atmospliere,  we  have 

and  hence  for  the  cylinder  volume 

i^^=:  0.888  —  . 
n 

For  an  engine  in  which  heat  was  added  under  constant  press- 
ure the  greatest  pressure  is  given  by 

T  ~\p) 
For  the  maximum  delivery  T-^  =  395.51  and  7^  =  273,  hence 

0.290T/ 

Pi  =  iJ     070    =  3.581  atmospheres. 
The  cylinder  volume  in  this  case  is 

i^^=  0.9107— , 

where  N  is  the  horse-power  and  p  =  1. 

While,  then,  the  volume  for  the  same  delivery  is  only  a  little 
greater,  the  greatest  pressure  is  much  less  than  in  the  first 
case. 

•Since  it  is  difficult  to  construct  engines  for  20  atmospheres' 
pressure,  we  cannot  use  the  system  which  gives  the  absolute 
maximum  of  work,  that  is,  gives  the  greatest  effect  with  the 
least  expenditure  of  fuel.  If  we  run  the  pressure  up  to  14  or 
15  atmospheres,  the  cylinder  volume  will  be  much  greater  than 
for  the  other  systems. 

Formulce  for  Hot- Air  Engines — Shortest  Form. — The  formula 
for  the  absolute  maximum  delivery 

which  we  have  found  for  the  simple  cycle  process,  can  be  de- 
duced in  this  form  for  every  hot-air  engine. 


252  THERMODYNAMICS. 

Since  in  the  equation 

L^~[_s{T,-T,)-siT,-T)]     page  244, 

s  {T^  —  Ti)  is  the  heat  imparted,  and  s  {T^-T)  that  abstracted, 
of  which  the  first  is  denoted  by  Q  and  the  second  by  Qi,  we 
have 

But  now  we  have  proved  that  for  all  hot-air  engines 

when  the  heat  addition  or  abstraction  takes  place  according  to 
the  law 


_p-i 

v^=p 

rn^n  , 

=  etc. 

Hence 

n- 

T,:n- 

T  = 

T,:  T, 

3     or     = 

T,: 

T. 

Therefore 

s{T^ 

-T,):s^ 

[n- 

T)  = 

T2:T,= 

T, 

•T, 

or 

Q'.Q. 

=  T, 

■•T,= 

--  T,  :  T. 

Accordingly 

Q  = 

T 

Qx- 

QT 

or 

Q^ 

Q.'- 

Inserting  the  first  value  of  Q^  in  the  above  equation  for  L, 
we  have 

^  =  f  (^1  -  ^^^)  =  -§^{T,-T)  .  (LXIV:) 

Hence  we  can  calculate  the  delivery  of  a  hot-air  engine  from 
the  heat  imparted  QG,  the  initial  temperature  T,  and  the  tem- 
perature which  the  air  receives  by  compression  according  to 
the  adiabatic  curve. 


HOT-AIR  UJ^aiNE  WITH  COMPLETE  CYCLE  PROCESS.      253 

"We  see  that  for  G  =  1,  tliat  is,  for  one  kilogram  of  air,  tliis 
equation  is  identical  with  that  found  for  the  simple  cycle  pro- 
cess, Chapter  YI, 

If  we  insert  the  second  value  of  Qi,  we  have 

Therefore  we  can  find  the  delivery  from  the  heat  imparted 
QG,  the  temperature  T^  which  the  air  receives  in  the  heating 
apparatus,  and  that  which  it  has  after  expansion  in  the  working 
cylinder. 

Since  the  specific  heat  does  not  occur  in  these  formulae,  we 
see  that  the  delivery  of  a  hot-air  engine  is  independent  of  the 
specific  heat.  If,  for  example,  in  the  two  systems  already 
noticed,  Ericsson's  and  Sterling's,  the  temperatures  2\  and  T, 
or  To  and  7g,  as  well  as  the  amount  of  heat  QG,  are  the  same, 
the  engines  will  all  give  the  same  delivery.  The  weight  of  air 
in  the  Sterling  engine  must  indeed  be  greater  than  in  the  other, 
because  the  specific  heat  s  is  less.  Neither  has,  then,  any  ad- 
vantage over  the  other,  apart  from  the  dimensions  of  the  engine. 
The  delivery  of  both,  as  well  as  of  all  systems  in  which  there  is 
a  cycle  of  the  kind  in  question,  is  proportional  to  the  heat  im- 
parted and  to  the  temperatures  occurring  in  the  engine  only. 

We  can  also  find  the  delivery  L  from  the  heat  Q^  G  abstracted. 

If  we  insert  in  the  equation  for  L,  the  second  value  for  Q,  we 
have 


G_  ( Q1T2  _  r)\  __  Q\G (T  —T\ 


(LXVI.) 


If  we  insert  the  second  value  of  G,  we  have 


There  only  remains  to  deduce  formulfe  for  hot-air  engines 
in  which  the  compression  and  expansion  do  not  take  place 
according  to  the  adiabatic  curve,  but  according  to  some  other, 
for  which  the  general  law  is 


254  THEBMODYNAMIGS. 

where  —  possesses  a  different  volume  from  that  for  the  other 

curves  of  heat  addition  and  subtraction.  The  deduction  of 
such  formulae  is  unnecessary,  since  they  are  applicable  to  no 
existing  systems,  and  it  is  improbable  that  in  future  any  engines 
will  be  constructed  to  which  they  are  applicable. 

As  already  remarked,  our  discussion  and  formulae  do  not 
apply  to  the  more  recent  engines  of  Laubereau  or  Lehmann,  in 
which  the  air  does  not  go  through  a  simple  cycle  process,  and 
which,  according  to  the  author's  view  (particularly  the  latter), 
are  especially  suited  for  minor  industrial  uses.  We  shall  there- 
fore seek  at  the  close  of  this  chapter  to  briefly  describe  a  hot- 
air  engine,  the  principle  of  which  seems,  from  a  practical  stand- 
point, worthy  of  notice.  We  allude  to  the  high-pressure  engine 
of  Eichard  linger,  described  in  the  Civil- Ingenieur  and  FolytecJm. 
Journal  In  the  next  chapter  we  will  treat  in  detail  of  the 
engines  of  Laubereau  and  Lehmann. 

Construction  of  anger's  Engine. — In  the  hot-air  engines  con- 
sidered in  this  chapter,  the  cold  air  is  first  compressed  adia- 
batically  in  the  feed-cylinder,  and  its  tension  increases,  there- 
fore, according  to  the  exponential  law  of  Mariotte.  If,  how- 
ever, we  compress  the  air  in  the  feed-pump  while  we  abstract 
the  heat  developed,  the  expansive  force  increases  according  to 
the  simple  law  of  Mariotte,  and  the  compressed  air  posesses  at 
the  same  pressure  pi,  a  much  less  volume  than  when  the  com- 
pression took  place  adiabatically.  Since,  also,  the  temperature 
of  this  air  is  less,  it  will  expand  much  more  when  lieated  to  the 
same  degree,  and  hence  perform  more  work ;  or  inversely,  for 
the  same  performance  it  expands  less,  and  hence  the  volume  of 
the  working  cylinder  is  less.  Also,  the  same  weight  of  air  is 
heated  to  a  less  degree  for  the  same  amount  of  heat  imparted, 
and  hence  the  highest  temperature  in  the  engine  is  less  than 
when  the  air  is  compressed,  according  to  the  exponential  law 
of  Mariotte. 

This  principle  has  been  applied  by  Eichard  linger,  in  his 
high-pressure  caloric  engine.  The  compressed  air  in  the  feed- 
pump is  cooled  by  injecting  cold  water  in  spray,  which  is  thus 
converted  into  steam.  This  water  is  forced  into  the  feed-cylin- 
der by  a  small  pump,  worked  by  the  engine  itself.     In  this 


VNGER8  ENGINE.  255 

way,  according  to  the  Journal,  the  air  is  brought  to  about  30'' 
C,  for  a  pressure  of  6  atmospheres  (5  atmospheres  effective). 
The  steam  formed  increases  somewhat  the  expansive  force  of 
the  air,  and  probably  diminishes  the  piston  friction. 

The  furnace  is  a  cylindrical  space,  inclosed  by  iron  plates, 
spherical  above  and  below.  It  consists  of  three  annular  con- 
centric spaces.  In  the  central  space  the  coal  is  consumed. 
This  is  connected  by  openings  with  the  inner  space,  which  is 
closed  above,  but  open  below.  This  latter  communicates  with 
the  outer  space. 

Into  the  fireplace  proper,  as  well  as  into  the  inner  space,  a 
part  of  the  compressed  air  is  forced  by  the  feed-pump  by  nar- 
row pipes,  thus  securing  perfect  combustion.  The  hot  pro- 
ducts of  combustion  then  mix  with  the  larger  part  of  the  cold 
air  furnished  by  the  compression-pump,  which  fills  the  outer 
annular  space,  where  it  is  heated  by  the  central  fire  space.  In 
this  way  the  air  receives  a  temperature  of  250  to  300". 

The  hot  air  is  then  led  by  special  pipes  to  the  valve  chests, 
and  enters  first  on  one,  then  on  the  other  side  of  the  piston. 

Before  the  compressed  air  reaches  the  furnace  it  enters  a 
receiver,  probably  in  order  better  to  regulate  the  air  necessary 
for  combustion.  {Dingier  s  Folytechn.  Journal,  Bd.  clxxxvi., 
Heft  1.) 


CHAPTEK  X. 

THE   HOT-AIR  ENGINES   OF  LAUBEREAU  AND  LEHMANN. 

In  the  preceding  chapters  we  have  given  the  theory  of  those 
open  and  closed  hot-air  engines  in  which  a  definite  volume  of 
air  makes  in  the  engine  a  cycle  process.  If  the  engine  is  sin- 
gle-acting, as,  for  example,  Ericsson's,  such  a  cycle  is  com- 
pleted during  one  revolution ;  in  double-acting  engines  we  have 
also  a  cycle  during  the  same  period,  but,  other  things  being 
the  same,  we  have  a  double  weight  of  air,  so  that  for  the  same 
dimensions  we  have  a  double  performance.  The  two  engines 
which  we  now  consider,  are  in  reality  closed  engines,  but  the 
air  in  them  does  not  complete  a  cycle  in  the  loay  heretofore  as- 
sumed. Thus,  while  heretofore  the  entire  inclosed  air  volume 
was  either  compressed  or  rarefied,  now  only  a  part  is  thus 
treated,  while  the  other  part  is  in  another  condition.  For  this 
reason  the  formulae  thus  far  developed  do  not  apply  to  these 
engines.  We  cannot,  therefore,  determine  their  performance 
from  the  heat  added  or  abstracted  and  the  temperature  fall, 
according  to  the  fundamental  principles  of  the  mechanical 
theory  of  heat,  but  we  must  rather  adopt  the  method  which 
we  have  followed  in  the  calculation  of  the  Ericsson  engine. 

Description  of  the  Lauhereau  Engine. — This  engine  consists  of 
a  hollow  iron  cylinder,  abed  (see  Eigs.  41  and  42,  following), 
surrounded  by  a  somewhat  wider  cylinder,  efgh.  The  space 
between  is  filled  with  cold  water,  in  order  to  cool  the  air  in 
the  cylinder.  In  the  lower  part  of  the  cylinder  is  a  bell,  of 
cast-iron,  whose  sides  are  corrugated,  in  order  to  afford  a 
greater  surface  to  the  hot  air.  The  sides  of  this  bell,  as  well 
as  of  the  cylinder,  are  shaded  black  in  the  Figs.  41  and  42.     In 


HOT-AIR  ENGINE  OF  LAUBEEEAU. 


257 


Fig.  42  tlie  bell  is  lieatecl  by  gas,  brouglit  ou  tlirougii  tlie  pipe  o 
and  burner  I.  To  secure  tbe  air  necessary  for  active  combus- 
tion, I  is  surrounded  by  a  tube,  mn,  tlirougli  wliicli  a  current 
of  air  passes.  In 
Fig.  41  tlie  fur- 
nace A  is  su23plied 
with  coal  by  tlie 
cast-iron  door  p, 
the  products  of 
combustion  pass 
out  tlirougli  m,  and 
thus  the  bell  ih 
is  heated.  These 
products  collect 
at  q,  and  pass  off 
through  the  chim- 
ney ss. 

The  interior  of 
the  cylinder  ahcd 
is  partly  filled  by 
iYiQ  distribution  pis- 
ton V,  the  exterior 
of  which  is  of  cast- 
steel,  and  which 
consists  of  two 
parts.  The  inte- 
rior of  both  parts,  and  the  space  between,  is  filled  with  some 
poor  conductor  of  heat.  The  piston  stock  has  a  rod  t,  which 
passes  air  tight  through  a  stuffing-box  in  the  cover  of  the  cyl- 
inder ahcd.  The  continuation  of  the  distribution  piston  below 
forms  a  thin  plate  cylinder,  as  in  Ericsson's  engine,  which,  when 
the  piston  is  in  its  lowest  position,  incloses  the  bell  ik,  and 
easily  absorbs  its  radiant  heat. 

The  object  of  the  distribution  piston  is  as  follows.  Sup- 
pose a  certain  amount  of  air  in  the  cylinder  ahcd,  then  by  far 
the  greatest  portion  will  be  above  the  piston,  since  there  is  the 
greatest  free  space.  But  since  the  upper  part  of  the  cylinder 
ahcd  is  surrounded  by  cold  water,  the  air  there  must  have  the 
temperature  of  this  water.  If,  now,  the  piston  V  rises,  the 
air  must  j)ass  from  the  upper  to  the  lower  space,  and  come  in 
17 


25^ 


THEBMOD  YNAMIG8. 


contact  with  the  hot  plates  of  the  piston,  and  the  hot  bell  ih, 
so  that  it  is  immediately  heated,  and  has  a  great  tension.     If 

the  piston  again  de- 
scends, the  heated  air 
passes  into  the  upper 
cold  space  and  im- 
parts its  heat  to  the 
cold  water.  The  tem- 
perature thus  sinks  to 
its  original  value.  It 
follows  that  the  water 
surrounding  the  up- 
per part  of  the  cylin- 
der must  be  constant- 
ly renewed.  This  is 
done  by  a  small  pump, 
P,  merely  indicated 
in  the  Figures. 

Now  that  we  have 
seen  how  the  air  in  the 
engine  is  alternately 
heated  and  cooled, 
and  thus  has  a  greater 
and  less  expansive 
force,  let  us  explain 
how  the  increased  force  of  the  heated  air  is  converted  into 
mechanical  work. 

From  the  lower  part  of  the  inner  space  of  abed,  in  the  neigh- 
borhood of  li,  a  pipe,  dotted  in  Fig.  42,  leads  to  the  lower  part 
of  the  working  cylinder  B.  This  cylinder  is  open  above.  In 
the  bottom  is  a  cock,  h^,  by  which  air  can  be  admitted  to  the 
cylinder  and  to  the  engine.  The  cylinder  is  fitted  with  an  air- 
tight piston,  K,  and  rod,  Jc^,  which  works  vertically  up  and  down 
through  fixed  guides  above.  By  the  forked-shaped  connect- 
ing-rod g-i,  the  crank  vvi  and  shaft  iviv  are  set  in  motion. 
The  shaft  passes  through  boxes  l^li,  borne  by  the  frame  CC. 
Upon  the  shaft  is  the  fly-wheel  SS,  and  the  disc  r,  which  by 
means  of  the  belt  r^  and  the  pulley  r^  work  the  pump  P.  The 
shaft  iviu  is  divided  in  the  middle,  and  furnished  with  two  small 
cranks  zz,  connected  by  a  triangular  cam  g^,  Fig.  41.     This  cam 


HOT-AIB  ENGINE  OF  LAUBEREAU.  259 

moves  in  the  rectangular  frame  uu,  wliicli  is  connected  below 
witli  the  rod  t  of  the  distribution  piston.  Above  it  is  connected 
to  two  rods,  /g  and  ^3,  which  work  A^ertically  in  the  guides  /i/i. 
The  cam  g^  thus  answers  to  the  eccentric  in  the  steam  engine. 

When,  now,  the  engine  is  put  in  motion,  the  gas  is  either 
kindled  or  burning  coal  introduced  through  the  door  p.  After 
a  few  minutes,  when  the  bell  ik  is  sufficiently  heated,  the  engine 
is  turned  by  means  of  the  fly-wheel  SS  beyond  its  dead  point, 
so  that  the  distribution  piston  Fis  moved  upwards.  The  cold 
air  is  thus  forced  rapidly  into  the  lower  warm  space,  where  it 
is  heated  up  to  about  300°  C  The  tension,  before  1  atmos- 
phere, thus  becomes  about  1|  or  more,  according  to  the  con- 
struction of  the  machine.  By  reason  of  this  increased  tension 
the  piston  h  in  the  working  cylinder  is  raised.  When  the  dis- 
tribution piston  has  reached  the  upper  end  of  its  stroke,  the 
working  piston  has  still  some  distance  to  go.  The  first,  there- 
fore, descends  while  the  working  piston  completes  its  stroke. 
The  tension  of  the  air  in  the  engine  sinks  immediately  below  1 
atmosphere,  and  the  working  piston  is  forced  down  by  the 
outer  air  pressure. 

At  the  moment  when  the  latter  commences  to  ascend,  the 
distribution  piston  has  already  risen  part  way,  a  part  of  the 
cold  air  is  again  heated,  the  entire  air  mass  has  thus  a  greater 
tension,  and  the  working  piston  is  again  raised,  and  so  on. 

While  the  motion  of  the  working  piston  is  tolerably  uniform, 
that  of  the  distribution  piston  is  more  irregular.  At  the  upper 
and  lower  points  of  its  stroke,  it  must  linger,  while  the  stroke 
up  and  down  is  quickly  accomplished.  Indeed,  it  may  even  be 
that  the  distribution  piston  makes  its  entire  stroke  while  the 
working  piston  is  reversing  its  motion.  The  motion  of  the  dis- 
tribution piston  is  dependent  upon  the  construction  and  shape 
of  the  triangular  cam  g.^  and  frame  uu.  (Such  an  arrangement 
is  represented  in  Fig.  777,  Art.  475,  of  Du  Bois'  translation  of 
Weisbach's  Mechanics  of  Engineering,  vol.  2.) 
,  According  to  reports,  these  engines  of  Laubereau  should  not 
make  less  than  500  revolutions  per  minute,  and  just  so  often 
the  air  must  be  heated  and  cooled.  From  a  practical  stand- 
point this  appears  hardly  possible.  If  the  data  are  correct, 
we  see  how  very  rapidly  air  can  receive  and  give  up  heat,  when 
made  to  move  over  warm  or  cold  plates. 


260  THERMODYNAMICS. 

Theory  of  Lauhereaiis  Engine. — Let  us  first  investigate  tlie 
tension  relations  in  tlie  engine.  We  may  proceed  here  in  a 
manner  similar  to  that  adopted  for  the  Ericsson  engine.  We 
use  the  following  notation. 

First,  it  is  assumed  that  the  cylinder  efgli,  as  well  as  the  dis- 
tribution piston  V,  are  closed  above  and  below  by  plane  sur- 
faces.    Let 

F  sq.  meters  be  the  cross-section  of  the  distribution  piston,  or 
the  area  of  the  cylinder  abed. 

H  meters,  the  length  of  stroke  of  the  distribution  piston. 

Hi  meters,  that  distance  which  the  distribution  piston  passes 
over  while  the  working  piston  reverses  its  motion. 

h  meters,  the  stroke  of  the  working  piston. 

Ih  meters,  the  distance  passed  over  while  the  distribution  pis- 
ton passes  through  H  —  H^. 

G  the  weight  of  air  in  the  engine,  which  is  always  constant. 

T  the  absolute  temperature  of  the  cold  air. 

t  its  temperature  Centigrade. 

Ti  the  highest  absolute  temperature  of  the  air. 

]j  the  pressure  of  the  cold  air  in  the  cylinder  dbcd,  when  the 
distribution  piston  is  below. 

^1  the  highest,  and 

2^i  the  lowest  pressure  during  a  revolution.     Finally 

h  meters,  the  height  of  the  prejudicial  space  or  clearance,  con- 
sidered not  as  annular  but  as  a  cylinder. 

If  the  distribution  piston  is  below,  the  air  volume  above  it  is 

FH 

of  the  temperature  T  and  pressure  p. 

This  weighs,  according  to  known  principles, 

-^  kilograms. 

Below  the  piston  we  have  the  air  volume 

Fh 

of  the  pressure  ]}  and  temperature  T^ ,  whose  weight  is 

F^ 
BTi' 


HOT- Am  ENGINE  OF  LAUBEREAU.  261 

Tlie  entire  weiglit  of  air  inclosed  is  tlien 

^      RT  ^  RT^       \T^  Tj    R     '    '    '     ^^'' 

The  distribution  piston  lias  passed  tlirongh  the  distance 
H  —  Hi  when  the  tension  in  the  entire  engine  has  risen  to  2h' 
Above  the  piston  there  is  now  the  air  weight 

FH.io, 
RT  ' 

and  below 

F{H-Hi)2h      Fhp, 
RTi  RTi  ' 

so  that  for  this  position  the  air  weight  is 

r._FH,ix  ,    {H-H,  +  h)Fp, 
^       BT    '^  BT, 

_rH,      H-H,  +  h-]Fp, 

From  (1)  and  (2)  we  have 

2h  _  3Ti  +  ^T 

p  ~  H,Ti  +  {H-H,  +  }))T   '     '    • 


(3). 


If  we  put  H—  Ri  =  H,  OT  Hi  =  0,  that  is,  if  we  assume  the 
working  piston  to  be  first  raised  when  the  distribution  piston 
is  at  the  upper  end  of  its  stroke,  we  have 

20^  ^  HTi  +  IT 

p       {H+-b)T' 

This  is  evidently  the  greatest  value  which  2h  can  have. 
Since  we  can  also  write 

HT,  +  hT  .,. 

^^  =  {H+h)TP ^^)' 

we  see  that_^:)i  depends  essentially  upon  the  initial  pressure  7:1. 
If  we  have  6  =  0,  we  have 

P.  =  ^P    .    .     .    =    ^    =    =     (5). 


262  THEBM0DYNAMIC8. 

Therefore  tlie  pressure  p^^  is  directly  as  tlie  temperature  7\ 
and  the  initial  pressure  p,  and  inversely  as  the  temperature  T. 

After  the  distribution  piston  has  risen  the  distance  H^,  the 
working  piston  moves  through  the  distance  hi  under  the  con- 
stant pressure  2^x-     The  air  weight  is  now 

From  (6)  and  (2)  we  have 

A  = ^-^ '--     and 

K^^    ^{T,-T) (7). 

If  again  we  put  H,  =  0,  we  shall  have  li,  =  0,  that  is,  the 
working  piston  goes  through  no  distance  under  constant  press- 
ure, but  is  raised  by  expansion  of  the  air. 

If  further  we  take/  =  F and  take  T^  =  2  x  273  and  ^  =  273, 
that  is,  if  we  warm  the  air  from  0  to  273°,  we  have,  as  is  evident. 

Let  us  determine  now  the  height  li,  which  the  working  piston 
has  to  move,  in  order  that  the  pressure  may  sink  to  the  origi- 
nal pressure  ^j. 

When  the  working  piston  has  passed  through  li,  the  air  vol- 
ume in  the  engine  is 

FH+Fh+flK 

This  volume,  at  the  pressure  p  and  temperature  T,,  weighs 

^  FBp^   ,Fhpfhp_(H_h\pfkp^ 
RT,   ^  RT,RT,~~\T,       Tj'B^  BT,' 

Equating  this  to  (1),  we  obtain 


FH 

+  Fb  +  fJi 
BT, 

_FH 
~  BT  ^ 

Fh 
BT, 

after 

reduction 

7       ^^, 

[T,-T). 

. 

(8). 


HOT-AIR  ENGINE  OF  LA  VBEBEA  U.  263 

From  (7)  and  (8)  we  liave  now 

While  now  tlie  working  piston  is  at  its  highest  point,  the 
distribution  piston  passes  through  H—  H^,  and  the  pressure 
of  the  air  sinks  from  p  to  ^ji.     The  inclosed  air  weight  is  now 

^_fhjp,^  FH^       Fhp,      F{H-H,)p, 
RT,        R2\    "^  R'T^  RT 

Equating  this  to  (1),  we  have 

(2^-  H,)i%T^  +  (ZTi  +  6  -  H)p.T=  (HT,  +  IT) p 


2h_ HT.  +  hT 

p  ~  {^H-H,)  T^  -  {H-  H,-h)T' 

If  here  we  put  H—  H^^  —  H,  or  H^  —  0,  we  have 

p,_         HT,  +  hT 

p  ~2HT,-{H-h)T  '    '    ' 


(9). 


(10). 


EXAMPLE. 

Let  -F'  =  1  sq.  meter,  /=  ^sq.  meter,  //=:  0.3  m.,  and  Hi  =0.1  meter.    Also, 
J  =  0.03  meter,  the  temperature  t  of  the  cold  air  0°,  or  T=  373,  and  of  the  hot 
air  i,  =  373%  or  T^  =2  x  373.     What  are  the  pressure  relations  in  the  engine, 
and  how  great  are  h^  and  h  ? 
From  (3), 

^_  0.2  X  3  X  373  +  0.08  x  878 

i?    ~  0.1  X  3  X  873  +  (0.3  -  0.1  +  0.03)373 

0.4  +  0.03       0.43       ,  010    +         T, 
=  a3T(U3  =  033  =  ^'^^^  atmospheres. 

If  Hi  were  0,  we  have  from  (4) 

p,       0.3  X  3  X  373  +  0.03  x  373       0.43       .  ^^^    ,  , 

y  =      0.3x373  +  0.03x373      =  0:33  =  ^'^^^  atmospheres. 

This  is  evidently  the  highest  pressure  which  can  be  attained  for  the  tempera- 
ture 373°,  when  only  air  is  inclosed  in  the  engine. 
If  the  prejudicial  space  is  zero,  we  have 

^ ,  =  3  atmospheres. 

The  minimum  pressure  jSa  is  from  (9) 

Ps  0.3  x  8  X  373  +  0.03  x  373  0.43 


p        (0.4  -  0.1)3  X  373  -  (0.3  -  0.1  -  0.03)373  ~  0.53 


=  0.808  atmosphere. 


264  THERMODYNAMICS. 

If  again  here  H^  =  0,  that  is,  if  the  distribution  piston  completes  its  stroke 
while  the  working  piston  lingers  at  the  upper  end  of  its  stroke,  we  have  from  (10) 

«3  0.2  X  2  X  273  +  0.02  x  273  0.42      ^  ^^„    ^ 

Y  =  2  X  0.2x2x278 -(0.2 -0.02)  273  =  0:62  =  ^'^'^  atmosphere. 

From  this  example  we  see  very  plainly  that  in  the  construction  of  the  engine 
care  should  be  taken  to  make  the  motion  of  the  distribution  piston  such  that  it 
shall  pass  through  its  entire  stroke  while  the  "working  piston  lingers  at  the  dead 
points,  and  that  it  shall  linger  on  its  own  dead  points  until  the  working  piston 
has  completed  its  entire  stroke.  Finally,  the  prejudicial  space  or  clearance 
must  be  as  small  as  possible. 

We  see,  further,  that  only  that  portion  of  the  inclosed  air  performs  outer 
work  by  which  the  air  volume  is  increased  by  heating.  The  entire  air  volume 
contained  in  the  distribution  cylinder  takes  no  part  in  the  performance  of  work. 
Since  it  is  heated,  however,  and  hence  its  inner  work  increased,  that  heat,  neces- 
sary for  the  increase  of  inner  work,  is  lost.  The  engine  works,  therefore,  less 
advantageously  than  those  hot-air  engines  considered  in  preceding  chapters. 

It  remains  to  determine  from  our  formulte  the  entire  stroke  h,  of  the  working 
piston,  and  that  part  of  it  which  the  piston  describes  under  constant  pressure. 
We  can  then  calculate  the  performance  of  the  engine. 

From  (7), 

h,  =  \    |il  (2  X  273  -  273)  =  0.2  meters. 

The  entire  stroke  h  is  from  (8), 

^^  =  ^    ^  (2  X  273  -  273)  =  0.4  metres. 

We  have,  therefore,  as  already  proved, 

A,  :h  =  H^:H. 

.  Delivery  of  the  Engine. — In  order  to  calculate  the  delivery  of 
tlie  engine,  we  proceed,  on  account  of  the  small  differences  of 
tension,  in  the  following  manner  : 

The  delivery  L^,  under  constant  pressure,  if  p  is  the  outer 
air  pressure,  is 

Li=f{Pi-p)lh. 

If  we  assume  the  mean  pressure  during  expansion  at  , 

A 

which  varies  but  little  from  the  actual,  we  have  for  the  deliv- 
ery during  expansion 

Hence  the  entire  delivery  during  the  rise  of  the  piston  is 


HOT-AIR  ENGINE  OF  LA  UBEBEA  U.  265 

When  tlie  working  piston  falls,  it  describes,  under  the  con- 
stant pressure^:*— ^2,  the  distance  Ih.  The  work  of  the  outer 
air  pressure  is  hence, 

From  here  on,  the  pressure  diminishes  gradually  down  to 
zero. 

We  have,  therefore,  for  the  delivery  during  the  descent 
h  —  h]_, 

Hence  the  delivery  during  the  descent  of  the  piston,  due  to 
the  pressure  of  the  atmosphere,  is 

fh{p-p.)+f{h-lh)^^=f{li  +  lh)^^^    .     (2). 


The  entire  delivery  of  the  engine  per  revolution  is  hence 
L=f(h  +  h)  ^^^  +f(h  +  h)  ^^^ 

=/(/>  + /O^^. 

If  the  engine  makes  per  minute  n  revolutions,  and  id  is  the 
efficiency,  the  delivery  per  second  is 

A  =  10334  ^  ^/  (^  +  ^0  -^^'2^^'  meter-kilograms, 

and  the  delivery  in  horse-power  is 

,^      10334        ./.   ,  7  ^P\-P% 

=  2.296m./(A  +  A,)^^^    •     (LXYIIL) 
If  we  take  the  preceding  example  and  assume  that  the  engine 


266  THERMODYNAMICS. 

makes  45  revolutions  per  minute,  and  lias  an  efficiency  of  0.5, 
we  have 

iV=  2.296  X  45  X  0.5  x  0.5  (0.4  +  0.2)  ^'^^^  ~  ^'^^^ 


=  4  liorse-powers. 

Tresca  has  observed  in  engines  of  this  kind,  for  cylinder 
diameter  of  0.5  meter,  and  hence  cylinder  cross-section  of  0.196 
square  meter,  stroke  of  0.4  meter,  and  effective  pressure  dur- 
ing rise  of  piston  of  0.25  atmosphere,  a  delivery  of  0.8  horse- 
power. For  a  cross-section  of  working  cylinder  of  0.5  square 
meter,  as  in  our  example,  the  delivery  for  this  slight  pressure 

0  5 
would  be  ^  -.'  ,  X  0.8  =  2.04  horse-power.     If  we  consider  that 

the  pressure  in  our  example  is  0.313,  and  that  the  effective 
pressure  of  the  atmosphere  is  also  greater,  we  can  conclude 
that  our  formula  is  reliable  if  the  number  of  revolutions  is  not 
far  from  45. 

The  delivery  is  considerably  greater  when  pi  and  pa  have  the 
above  calculated  maximum  values,  or  when  the  engine  is  so 
constructed  that  the  working  piston  lingers  at  its  upj)er  or 
lower  positions  while  the  distribution  piston  describes  a  com- 
plete stroke. 

Dimensions  for  a  given  Delivery. — If  we  put  in  the  last  for- 
mula in  place  of  h  and  1^  the  values  from  (7)  and  (8),  we  have 

iV=  2.296;^^y  ^{T,-  T){H+  H,) ^L^    .    (LXIX.) 

whence  it  appears  that  the  delivery  increases  not  only  with 
the  area  F  and  the  height  H  of  the  distribution  cylinder,  but 
also  with  the  temperature  T^. 

From  this  formula  we  have  at  once  the  area  F  for  an  engine 
of  given  power.     Thus, 

2TiV 
^Z"  'M'dQnw{T,-T)  (ZT-f  H,){p,  -  p,)   '    (^^^'^ 

Here  n,  w,  T-^,  T,  and  H,  H^  are  considered  as  given.  The 
values  of^:>i  and  j:>2  are  then  found  from  (3)  and  (9). 


HOT-AIR  ENGINE  OF  LAUBEREAU.  267 

For  the  maximum  delivery,  wliere  H^  =  0,  we  have 

^"  2.396  nwH{T^  -  T)  {p^  -2h)   '    ^^^^^-^ 

EXAMPLE. 

What  must  F  and  /  be.  when  the  actual  delivery  N  of  the  engine  is  two  horse- 
power, and  IV  =  0.30,  n  =  60,  T^  =  2  x  273,  T=  273,  11=  0.05,  and  &  =  0.01  ? 
If  we  require  from  the  engine  the  maximum  of  work,  we  have  from  the  equation 

^i—  =         '^  by  msertnig  the  numerical  values. 

0.05  X  2  X  273  +  0.01  x  273      ,  ^oo    . 

(0.05  + 0.01)  X  273 =  1-^^^  atmospheres, 

and  from  Equation  10 

Pi-  ^r,  +  bH 

p  ^  2HT,  -  {H-1))T 

0.05  X  2  X  273  +  0.01  x  273  n  poo    ^ 

0.688  atmosphere. 


~  2  X  0.05  X  2  X  273  -  (0.05  -  0.01)  x  273 

Hence 

2  X  273  X  2 


3  X  296  X  60  X  0.3  x  0.05  x  273  x  1.145 

4 


_  Q        =  1.69  square  meters.* 

The  entire  volume  of  the  air  inclosed  in  the  distribution  cylinder  must  there- 
fore be 

1.69  x  0.05  r=  0.0845  cubic  meters. 

From  (8)  we  have  for  the  volume  of  the  working  cylinder 
2  V  27'-? 97^ 

fh  =  0.0845  273  "=  ^'^^^^  ^^^^^  meters. 

If  we  take  h  =  0.05  meters,  we  have  /==  -^— j —  =  0.169  sq.  meters. 


The  Hot- Air  Engine  of  Lehmann. — In  the  engine  of  Laubereau 
we  meet  with  two  evils,  the  direct  action  of  the  hot  air  upon 
the  working  piston,  and  the  alternate  heating  and  cooling  of  a 
part  of  the  inclosed  air  which  does  not  contribute  to  the  action 
of  the  machine.  The  first  objection  is  common  also  to  the  en- 
gines of  linger  and  Belou,  while  Ericsson  has  avoided  it  in  his 

*  If  \vc  take  for  the  stroke  O.iim.  inuteaclof  0.05m.,  we  should  have  for  F,  0.169  sqtiare  meters. 
Hence  the  radius  of  the  distribution  piston  would  be  onlj'  0.233  meter. 


268 


THERMOD  TNAMIG8. 


engine  in  a  very  ingenious  manner.  In  fact  this  invention  is 
specially  distinguished  in  that  the  working  piston  always  re- 
mains cool  (the  temperature  at  most  40^  to  50"^),  so  that  it  can 
be  Avell  lubricated,  and  also  in  that  the  motion  of  the  piston  is 
transmitted  in  a  very  simple  and  ingenious  manner  to  the  fly- 
wheel and  this  to  the  feed  piston. 

The  objections  to  Laube- 
reau's  engine  have  been  met  by 
Lehmann,  who,  with  the  same 
idea  at  bottom,  has  joined  the 
special  advantages  of  Erics- 
son's invention.  In  Lehmann's 
hot-air  engine,  also,  an  inclosed 
quantity  of  air  is  alternately 
heated  and  cooled,  but  it  does 
not  act  directly  upon  the  work- 
ing piston.  This  latter  is  pro- 
tected by  the  feed  piston  from 
the  heat  of  radiation  and  con- 
duction, while  still,  the  expan- 
sive force  of  the  hot  air  is  ap- 
plied in  the  same  cylinder  in 
which  it  is  heated  and  cooled. 
For  this  reason,  a  part  of  the 
air  is  not  heated  and  cooled 
unnecessarily  at  every  revolu- 
tion. 

Fig.  43  represents  a  section 
of  Lehmann's  engine,  a  de- 
scription of  which,  together 
with  a  number  of  experi- 
mental results,  have  been 
given  byEckerth  in  the  Viertel- 
jalirssclirift  des  deutsclien  Inge- 
nieur-und  ArchiteJcten-  Vereins,  1 
Jahrgang,  2  Heft. 

ABGD,  Fig.  43,  is  a  horizon- 
tal cylinder  of  cast-iron,  open 
in  front  and  closed  behind.     S  is  the  iire  space,  with  the  ash 
pit  T.     The  hot  air  plays  around  the  outside  of  the  cylinder  at 


HOT-AIR  ENGINE  OF  LEIIMANN.  269 

the  left  end,  and  tlien  mounts  tlirougli  tlie  pipe  B,  whicli  is  fur- 
nished with  a  throttle  valve  for  regulating  the  draft,  into  the 
air.  About  fds  of  the  cylinder  is  surrounded  by  a  wider  cyl- 
inder, LMNP,  and  the  annular  space  between  is  kept  filled 
with  cold  water,  which  enters  below  and  escapes  through  the 
pipe  Q  after  it  has  cooled  the  air. 

In  the  cylinder  ABCD  two  pistons  move.  One,  UU,  is  the 
working  piston,  which  is  made  to  fit  air-tight  by  a  leather 
washer.  This  washer  is  so  constructed  that  it  allows  air  to 
enter  when  the  outer  air  pressure  is  greater  than  the  inner,  but 
hugs  only  the  tighter  and  prevents  exit  when  the  inner  air 
pressure  is  greater  than  the  outer.  The  outer  piston,  called 
the  compressor,  consists  chiefly  of  an  air-tight  riveted  plate 
cylinder  EFGH,  stiffened  inside  by  the  piston  KK,  and  closed 
in  front  by  the  wooden  piston  EF.  The  diameter  of  this  cyl- 
inder is  but  little  less  than  that  of  ABCD,  so  there  is  only  a 
narrow  space  between. 

The  compressor  is  connected  with"  a  rod  VW,  which  passes 
air-tight  through  the  working  piston.  This  latter  transmits  its 
motion  to  the  fly-wheel  in  a  manner  similar  to  Ericsson's  en- 
gine, and  the  motion  of  the  compressor  is  similarly  effected. 

In  order  to  preserve  its  motion  in  a  right  line,  guides  are 
fitted  to  the  cylinder  ABCD,  and  to  diminish  the  friction  we 
have  a  roller  at  0. 

The  cranks  and  connecting-rods  by  which  the  motion  of  the 
working  piston  is  communicated  to  the  compressor  are  so  ar- 
ranged that  the  crank  for  the  latter  is  65°  in  advance  of  that 
for  the  working  piston. 

In  the  vicinity  of  X  there  is  a  regulator,  which  opens  a  valve 
when  the  normal  action  is  exceeded. 

The  mutual  motions  of  the  working  piston  and  compressor 
are  rejoresented  in  Fig.  44. 

The  circles  above  are  the  crank  circles.  We  have  divided 
each  into  12  equal  parts.  When  the  crank  of  the  working  pis- 
ton has  gone  from  1  to  2,  the  working  piston  has  gone  through 
the  distance  2a,  and  simultaneously  the  feed  piston  has  gone 
through  the  distance  26.  When  the  crank  of  the  working  pis- 
ton reaches  3,  the  working  piston  has  passed  through  3«,  and 
the  feed  piston  through  36,  and  so  on. 

We  see  now  from  the  diagram  that  when  the  working  piston 


270 


TUERMOD  YNAMIC8. 


readies   its  extreme   position  on  tlie  right  (see  Fig.  43),  tlie 
compressor   lias   gone    almost   the  half   of  its    stroke  toward 

the  left.  This  posi- 
tion we  have  indi- 
cated in  the  diagram 
by  /.  The  space 
right  and  left  of  the 
comj)ressor  is  then 
equal,  but  in  the 
one  we  have  cold 
air  and  in  the  other 
hot  air,  while  the 
air  layer  in  the  nar- 
row annular  space 
between  compres- 
sor and  cylinder 
forms  a  poor  con- 
ductor, b  y  which 
the  propagation  of 
heat  from  the  hot 
to  the  cold  space  is 
prevented.  This  air 
answers,  then,  the 
purpose  of  the  ring 
valve  in  Ericsson's 
engine. 

When  the  two 
pistons  have  the 
position  in  /,  the 
compressor  goes  to- 
ward the  left  some- 
what more  rajoidly 
than  the  working 
piston.  The  hof 
air  is  thus  driven 
out  of  the  hot  space 
behind  the  c  o  m  - 
pressor  into  the 
cold  space  in  front, 
and  its   tension  is 


HOT-AIR  ENGINE  OF  LEHMANN.  271 

diminislied.  Since,  Liowever,  tlie  working  piston  also  moves 
toward  the  left,  the  inclosed  air  is  compressed.  But  the 
increase  of  the  tension  by  the  compression  is  at  first  less  than 
the  decrease  by  cooling.  If,  therefore,  the  entire  air  yolume 
had  at  first  the  pressure  of  the  atmosphere,  its  pressure  will 
be  now  somewhat  less.  In  Ericsson's  engine  the  same  is  true. 
As  soon  as  the  feed  piston  begins  to  move  backward,  the  air 
between  the  two  pistons  is  rarefied,  and  cold  air  enters  through 
the  valve  in  the  working  piston.  Eckerth,  however,  ascribes 
this  decrease  of  tension  to  losses  of  air.  Erom  3  on  (see  dia- 
gram), the  working  piston  and  compressor  move  with  almost 
equal  velocity  toward  the  left,  and  since  at  3  there  is  but  little 
hot  air  behind  the  compressor,  the  entire  inclosed  air  volume 
soon  takes  the  pressure  of  the  atmosphere,  as  the  closed  curiae 
LILIII.IV.  above  the  crank  circle  in  the  diagram  represents. 
This  is  obtained  by  laying  off  from  the  line  a [3  the  pressures 
corresponding  to  the  positions  I.ILIILIV.  Erom  4  on,  where 
the  working  piston  moves  toward  the  left  much  more  rapidly 
than  the  compressor,  and  the  latter  is  almost  at  the  end  of  its 
stroke,  the  pressure  rises  above  the  atmosphere. 

At  //.  the  compressor  has  its  extreme  left  position,  and  all 
the  hot  air  has  passed  into  the  cold  space,  and  since  the  work- 
ing piston  also  has  nearly  completed  its  stroke  toward  the 
left,  the  pressure  is  considerably  higher  than  the  atmosphere. 
The  maximum  pressure,  however,  is  not  yet  attained.  Erom  //, 
to  III.  it  must  increase  rapidly,  for  now  the  compressor  re- 
turns and  forces  cold  air  into  the  hot  space,  while  the  work- 
ing piston  still  is  going  toward  the  left  as  before.  About  at  III., 
then,  we  have  the  maximum  pressure. 

Between  III.  and  IV.  it  remains  tolerably  constant,  because 
both  pistons  are  moving  together  toward  the  right  with  nearly 
equal  velocity,  and  because  the  increase  by  the  heating  of  the 
cold  air  is  balanced  b}^  the  increase  of  volume. 

Erom  IV.  on,  the  working  piston  goes  more  rapidly  to  the 
right  than  the  compressor,  and  the  pressure  sinks  gradually 
until  it  is  again  1  atmosphere. 


272  THEBM0DTNAMIC8. 

Calculation  of  the  Pressures  in  Lehnamis  Engine. — Let 

F  =  the  cross-section  of  the  larger  cylinder. 

jPj  =  that  of  the  compressor. 

I,  the  length  of  compressor. 

a,  the  distance  between  pistons  in  the  position  I. 

Ori,  the  distance  from  end  of  compressor  to  that  of  cylinder. 

T,  the  absolute  temperature  in  the  cold  space. 

Ti,  that  in  the  annular  space  between  compressor  and  cylinder. 

21,  that  in  the  hot  space. 

For  the  inclosed  air  weight  G  we  have,  when  2^  is  the  press- 
ure for  the  position  I, 

Fap      {F~F,)lp      Fa,p 
If  we  put  (F  —  Fi)l=:  ~  Fa,  we  have 


Again  for  the  position  //.  (see  diagram),  let  h  be  the  distance 
between  pistons,  h^  that  between  cylinder  end  and  compressor 
end,  pi  the  pressure  for  this  position,  then 


^^=-^(^44)  ■  •  ■  •  (^)- 


From  (1)  and  (2)  we  obtain 

Pi  ^  T^lhTj^T^ 
!P~  h  a         &i  * 

If  for  position  III.  we  have  c,  Ci,  and  ^2  instead  of  h,  hi  and^i, 
we  obtain 


rh 

a 
T 

+ 

nTi 

p 

c 
T 

+ 

4- 

Cl 

HOT-AIR  ENGINE  OF  LEHMANN.  273 

For  position  IV.,  taking  ^g,  d,  and  d^  in  the  same  signification 
as  p2>  c,  and  q  for  ///.,  we  have 

a         a         <\ 
p        d         a  di 

The  engine  described  by  Eckerth  was  of  1  horse-power,  and 
had  the  following  dimensions  : 

Diameter  of  working  piston 0.349  meters. 

Stroke  "  "     0.175       " 

Diameter  of  compressor 0.342       " 

Stroke       "  "         0.244 

Length       "  "         1.527 

Area  of  working  piston F  =  0.0957  sq.  meters. 

Further,  if  we  take  the  dimensions  of  the  drawings  given  by 
Eckerth  in  his  description  of  Lehmann's  engine  as  approxi- 
mately correct, 

a  =  6.5"  (Vienna)      a^  =  6.375"        h  =  7.5"  b,  =  0.25" 

c  =  3.5"  ci  =  2.8"         *  d  =  0.25"        d,  =  8.25". 

In  the  experiments  the  cold  water  entered  with  a  tempera- 
ture of  30°  to  60°,  and  left  at  40°  to  70°.     The  cold  air  had  then 

70  -J-  40 
a  somewhat  higher  temperature  than  — ^ =  55°  on  the  av- 

erage.  If  we  take  it  at  60°,  r=  273  +  60  =  333°.  If  we  assume 
the  mean  temperature  of  the  air  in  the  annular  space  at  100°, 
and  that  in  the  hot  space  at  300°,  we  have  T^  —  373,  and  T^  = 
573. 

According  to  Eckerth,  n  is  about  =  3.  We  have  taken  for  n 
2.5. 

Hence  we  have 

1-1  .        6.5  6.375 

p_  ^    333  2.5  X  373       573    ^  0.03762  _  ^  ^^^ 

p  7^  6.5  0^25   ~  0.02993  ~ 

333  2.5  X  373      573 

18 


274  TEEBMODTNAMICS. 

Also  for  position  III. 

6.5  6.5  6.375 

jQa  _  333  "*"  2.5  x  373        573   _  0.03762  _  .,  ^^.„ 
p  ~  ^  6.5  2.8    ~  0.02237  ~ 

333      2.5  X  373  ^  573 


For  if: 


6.5    .        6.5       ^6.375 


p^  _  333      2.5  X  373       573   _  0.03762  _  ^  ^^ 
p  ~  025  6.5  8.25  "  0.02212  '* 

333  "^  2.5  X  373  "^  573 


Experiments  with  the  indicator  gave  as  tlie  maximum  press- 
ure an  excess  of  10  pounds,  or  an  effective  pressure  of  10  + 14.1  = 
24.1.     Hence 

^  =  ^  =  1.709, 
p       14.1 

a  value  wliich  agrees  very  closely  witli  our  calculation.  The 
pressure  relations  are  here,  therefore,  more  favoraHe  than  in 
Ericsson's  engine,  and  considerably  more  so  than  in  Laube- 
reau's  engine,  as  might  have  been  expected  from  a  considera- 
tion of  the  construction. 

Delivery  of  tJie  Engine. — According  to  the  diagram,  the  pressure 
of  the  air,  while  the  working  piston  passes  from  I.  to  4,  is 
tolerably  constant,  and  somewhat  less  than  that  of  the  atmos- 
phere. From  4  to  III.,  it  increases  about  as  the  volume  di- 
minishes, and  the  increase  follows  the  isothermal  curve  there- 
fore. From  ///.  to  IV.  again,  the  pressure  remains  tolerably 
constant,  and  then  from  IK  to  1  decreases  almost  exactly 
according  to  the  ordinates  of  a  straight  line,  that  is,  the  de- 
crease of  pressure  is  proportional  to  the  distance  traversed  by 
the  working  piston. 

"We  have  denoted  the  area  of  the  piston  by  F.  Let  the 
distance  passed  over  from  ///.  to  IV.  be  w,  then  the  work  of 
the  air  during  this  motion  of  the  piston  is 

Li  =  Fwp^. 


HOT-AIR  ENaiNE  OF  LEHMANN.  275 

From  IV.  to  1  the  pressure  is        ^       kilograms.     If  the  dis- 


tance is  W],  the  work  is 


The  work  of  the  heated  air  is  therefore 

Further,  let  the  pressure  from  I.  to  4  be  j^o  ^^  the  average, 
and  the  distance  lo^,  then,  since  jh  <  p,  that  is,  is  less  than  the 
atmospheric  pressure,  the  work  of  this  latter  is 

L^  =  Fiv.:,{p  -po). 
So  that  the  entire  delivery  is 

L^  +  L^  +  L^  =  F [ivps  +  w, 21^1.  +  -^2  (^  -i?o))- 
The  work  of  compression  from  4  to  III.  is 
^4  =  2.3026^2^2  log  ^ 

if  V2  is  the  least  volume  and^  the  greatest  pressure. 

The  first  occurs  when  the  working  piston  reaches  the  back 
end  of  its  stroke.    Instead  of  J^  we  can  put  Fx,  and  have  then, 

X4  =  2.3026i^a:i92  log  ^, 
so  that  the  effective  delivery  per  stroke  is 

L^  +  L^  +  L^-L^~L~f\wp^  +  w^  2?d^  +  Wz(p  -po) 

-2.3026^^92  log  ^]. 


276  THEBMODTNAMICS. 

If  tliere  are  n  strokes  per  minute,  tlie  delivery  per  second  in 
horse-powers  is 

N=  gQ^  F  \^vp,  +  IV,  P^  +  w,  {p-2%)  -  2.3026xp, log^^J, 

where  p^i  P>  etc.,  are  expressed  in  kilograms.     If  we  express  the 
pressures  in  atmospheres,  we  have 

N=  6^  ^  l''^''  ^  '''  ^^  +  ''' ^^'  ~^'^  "  2-3026a;i.,log|^], 
or 

N  =  2.296Fn  Itops  +  lo,  ^^^^  +  iv^  (p  -  ^^o) 

-  2.3026^^2  log  ^'1     ....     (LXXII.) 

EXAMPLE. 

For  the  engine  already  discussed,  F  =  0.0957  sq.  meters,  and  n,  on  an  aver- 
age, was  97.  If  the  distances  w-^iv-^,  etc.,  traversed  by  the  working  piston,  are 
taken  from  the  diagram,  we  have  iv  =  0.06™,  -io,  =  0.115,  and  iv^  =  about  0.06'"- 
What  is  N,  when  we  takers  =  1.71,  and  jo,,  =  0.97  atmospheres  ? 

Since  the  least  volume  v..  —  0.03125  cubic  meters,  x  =  0.233"-,  hence 

i\r=  2.396  X  0.0957  x  97  [0.06  x  1.71  +  0.115^^^^^  +  0.06  x  0.03  -  2.3036 


X  0.323  x  1683 log  1.681 

N=  21.814  [0.1026  +  0.1558  +  0.0018  -  0.1941] 
=  21.314  X  0.0661  =  1.4088  horse-power. 

A  mean  of  several  experiments  with  the  friction  brake  gave 

73  8 
the  actual  delivery  73.8  meter-kilograms,  or  -,rr^  =  0.984  horse- 

0  984 
power,  so  that  the  efficiency  is  about —^----  =0.69.     Accord- 

''  1.4088 

ing  to  Eckerth,  who  reckoned  the  theoretical  delivery  from 
diagrams,  the  efficiency  was  about  0.66.  Both  seem  somewhat 
too  great.  For  a  \  horse-power  caloric  engine  of  Ericsson,  the 
efficiency,  according  to  experiments  by  the  author  (Dingler's 
PolytecJm.  Journal,  Bd.  clxxix.)  was  0.40,  a  result  with  which 
Tresca's  experiments  closely  agreed.     It  does  not  seem  likely 


HOT-AIB  ENGINE  OF  LEEMANN.  277 

that  tlie  efficiency  of  the  present  construction  is  greater  by  26 
to  29  per  cent. 

The  above  formul£e  for  the  calculation  of  the  power  of  a 
Lehmann  engine  may,  according  to  the  determinations  and 
views  of  the  author,  be  replaced,  as  long  as  the  proportions 
are  unchanged,  by  the  following  simple  formula : 

Lg  =  0.163''  (fi  —  ^2)  Gr  meter-kilograms, 

where  L^  is  the  delivery  per  second,  n  the  number  of  revolu- 
tions per  minute,  fi  the  highest  and  ^2  the  lowest  temperatures 
of  the  inclosed  air,  and  G  the  weight  of  air.  This  last  can  be 
easily  calculated  fi-om  (1).  We  have  thus  G  =  0.02784  kilo- 
grams. This  formula  rests  upon  similar  reasoning  to  that  on 
page  222. 

Eckerth  states  that  the  machine  required  13.5  cubic  feet,  or 
a  weight  of  426.4  kilograms  of  cooling  water.  This  was  fur- 
nished by  a  small  pump  worked  by  the  engine,  and  forced  into 
the  cold  space,  from  whence,  after  being  heated,  it  flowed  back 
to  the  reservoir.  In  2  hours  this  water  showed  an  increase  of 
61  -  35  =  26°,  or  13°  per  hour.  The  heat  absorbed  by  the 
water  was  hence 

426.4  X  13  =  5543.2  heat  units  per  hour. 

The  hourly  delivery  was 

73.8  X  60  X  60  =  265680  meter-kilograms, 

corresponding  to       .         =  626.6  heat  units. 

Hence  the  heat  units  per  hour  imparted  to  the  air  are 

5543.2  +  626.6  =  6170. 

The  ratio  of  the  heat  transformed  into  work  to  that  absorbed 
by  the  cooling  water  is  therefore 

626.6   _  ^^ 
5543:2  -^•^^^' 

The  engine  required  per  hour  4.585  kilograms  of  coal,  whose 


278  THERMODYNAMICS. 

heating  value  was  estimated  at  3,500  heat  units.     The  coal  fur- 
nished therefore 

4585  X  3500  heat  units. 

Since  only  626.6  units  were  utilized,  the  ratio  of  the  heat 
transformed  into  mechanical  work  to  that  furnished  by  the  coal, 
or  the  thermal  effect,  is 

A  gQcr    'qHaa  =  0.039,  or  about  4  per  cent. 
4.585  X  3500  ^ 


The  efficiency  of  the  furnace  was 

6170 
4.585  X  3500 


=  0.38. 


The  difference  in  temperature  of  the  cooling  water  at  en- 
trance and  exit  was,  while  the  engine  was  working  at  about  1 
horse-power,  about  constant  and  9°  Centigrade. 

The  heat  units  per  hour  imparted  to  the  water  was,  as  above, 
5543.2,  hence  per  minute  92.39.  The  amount  of  water  used  per 
minute  is  then 

— ^ —  =  10.266  kilograms  =  10.266  cubic  decimeters. 

The  diameter  of  the  pump,  as  also  its  stroke,  was  about  2 
inches. 

The  following  table,  taken  also  from  the  Vierteljahresschrift, 
gives  a  comparison  between  the  hot-air  engines  of  Ericsson, 
Laubereau,  and  Lehmann : 

Diameter  of    Useful  Delivery  Consumption  of       Cooling  Water  per 

System.        Workins;  per  Efficiency.      Fuel  per  Horse-        Horse-power  per 

Piston.  Second.  power  per  Hour.  Hour. 

Ericsson.... 0.414  m.        1.77  H.  P.        0.46        4.13  Ml.  of  coal 

o  f  heating 
value  7,000- 
7,500. 

Laubereau..0.343  m.        0.8      ■             0.40       4.5-5  kil.  coke  20-30  cubic  feet 

of      heating  for  a  rise  of 

power  7,000-  temperature 

7,500.  of  17". 

Lehmann.. 0.239  m.        1                       0.66 (?)   4.6    kil.     hard  6|  cub.  ft.   for 

coal  of  heat-  rise  of   tem- 

ing    power  perature     of 

about  3, 500.  26°. 


EOT-AIB  ENGINE  OF  LEHMANN.  279 

According  to  tlie  views  of  the  author,  the  engines  described 
are  esi3ecially  suited  for  the  lesser  industries.  They  are,  for 
reasons  cited,  more  diirable  than  Laubereau's,  and  the  annoy- 
ing pounding  of  Ericsson's  is  entirely  avoided.  Moreover,  the 
consumption  of  fuel,  as  shown  in  the  table,  is  only  the  half  of 
that  for  the  other  two  systems.  In  this  respect  it  is  as  econo- 
mical as  the  best  steam  engines. 

Caloric  engines  are  also  perfectly  free  from  danger,  and  their 
attendance  and  management  is  much  simpler  than  for  the  steam 
engine. 


CHAPTEE  XI. 

GAS   ENGINES,  ESPECIALLY  THOSE   OF   OTTO  AND  LANGEN. 

Almost  at  the  same  time  as  the  introductipn  of  the  Ericsson 
engine  in  Europe,  the  gas  engine  was  invented  by  Lenoir  in 
Paris.  Let  us  consider  the  circumstances  which  gave  rise  to 
the  idea. 

It  is  known  that  when  the  electric  spark  is  led  through  a  mix- 
ture of  air  and  hydrogen,  there  is  a  combustion  of  the  latter  as 
it  unites  with  a  part  of  the  oxygen  of  the  air  to  form  steam.  By 
this  combustion  the  air,  as  well  as  the  products  of  combustion, 
are  heated,  and,  if  the  sides  of  the  containing  vessel  are  mov- 
able, there  will  be  expansion  and  increase  of  volume ;  if  not, 
there  will  be  a  considerable  increase  of  pressure.  The  same  is 
the  case  when  gas  is  kindled.  Such  facts,  which  were  un- 
doubtedly known  to  Lenoir,  may  have  suggested  to  him  the 
gas  engine. 

This  machine  is  of  simple  construction,  and  in  exterior  ap- 
pearance very  similar  to  a  steam  engine  with  horizontal  cylinder. 
The  principal  part  is  thus  a  hollow  cylinder  closed  at  both  ends, 
in  which  moves  an  air-tight  piston.  This  is  connected  by  means 
of  piston  and  connecting-rods  with  the  crank  of  a  fly-wheel,  from 
which  by  a  belt  the  power  can  be  taken  off. 

Let  the  piston  be  at  the  right  end  of  its  stroke.  By  turning 
the  fly-wheel  it  moves  from  right  to  left.  While  it  moves  in 
this  direction,  a  mixture  of  illuminating  gas  or  of  hydrogen 
and  air  enters  through  a  slide  valve,  worked  of  course  by  the 
engine.  When  a  certain  quantity  of  this  mixture  has  entered, 
the  slide  shuts  in  the  mixture  in  the  cylinder,  and  at  this 
moment  a  spark  from  an  induction  coil  kindles  the  gas  and 
causes  an  increase  of  pressure.  This  drives  the  piston  to  the 
other  end  of  its  stroke.     Arrived  at  the  left  end,  a  second  slide 

280 


0A8  ENGINES  OF  OTTO  AND  LANOEN.  281 

allows  the  products  of  combustion,  together  witli  the  accom- 
panying air,  to  escape,  while  the  fly-wheel,  by  reason  of  its  liv- 
ing force,  goes  on  and  moves  the  piston  in  the  opposite  direc- 
tion, from  left  to  right.  While  it  moves  in  this  direction  the 
mixture  again  enters,  and  is  kindled  as  before,  and  thus  the 
piston  is  driven  toward  the  right. 

The  action  is  therefore  very  similar  to  that  of  a  non-condens- 
ing expansion  steam  engine.  But  while  the  greatest  work  is 
performed  by  the  entering  steam,  the  work  of  the  entering  gas 
mixture  is  here  zero.  During  this  interval  the  friction  of  the 
engine  and  the  resistances  must  be  overcome  by  the  living  force 
of  the  fly-wheel. 

As  the  explosion  of  the  gas  is  instantaneous,  the  increase  of 
pressure  is  also  immediate.  But  since  the  piston  can  only 
gradually  yield,  there  is  at  the  moment  of  firing  a  shock  which 
is  not  only  hurtful  to  the  moving  parts,  but  which  cannot  con- 
tribute to  the  motion  of  the  piston  or  the  performance  of  work. 
Hence  an  amount  of  heat  corresponding  to  this  impact  is  ab- 
stracted from  the  gas  and  transferred  to  the  cylinder  and  pis- 
ton. These  parts  would  soon  become  very  hot  if  heat  were  not 
abstracted.  The  cylinder  is  therefore  surrounded  with  a  jacket 
filled  with  cold  water.  In  this  way  the  temperature  of  the 
cylinder  sides  is  kept  down  to  80'  or  90^.  It  is,  however,  evident 
that  the  performance  can  by  no  means  be  as  great  as  the  con- 
sumption of  gas  would  warrant. 

This  objection  to  the  Lenoir  engine,  viz.,  that  a  large  part  of 
the  power  is  lost  by  impact,  and  yet,  in  spite  of  cooling,  the 
piston  andstufiing  boxes,  etc.,  are  considerably  heated,  has  been 
met  by  Otto  and  Langen,  of  Cologne,  in  a  very  ingenious  way, 
and  thus  a  machine  produced  whose  power  for  the  same  gas 
consumption  is  much  greater.  We  shall  discuss  it  more  at 
length  later  on. 

In  order  to  diminish  the  heat,  Hugon  introduced  with  the 
gas  a  quantity  of  water,  which  absorbed  in  vaporizing  consid- 
erable heat.  The  expansive  force  is  thus  indeed  somewhat 
diminished,  but  the  moving  parts  suffer  less,  and  the  machine 
requires  less  repair  and  is  more  durable. 

In  Germany  the  Lenoir  gas  engine  has  dropped  out  of  sight, 
although  in  the  beginning  various  establishments  sought  its 
improvement,   and  not  indeed  without  practical  results.     In 


282  THEBM0DYNAMIG8. 

France,  however,  especially  in  Paris,  it  lias  been  applied  in 
the  building  of  houses  to  the  raising  of  building  materials,  and 
in  England  also  it  has  been  used  for  stone  sawing,  polishing, 
pumjDing,  printing,  etc. 

Gas  engines  are,  as  is  evident  from  the  foregoing,  properly 
nothing  more  than  hot-air  engines  with  inner  fire.  But  while 
in  the  latter  the  air  is  heated  by  coal  or  coke,  in  the  former 
it  is  heated  by  the  combustion  of  illuminating  or  hydrogen  gas, 
kindled  either  by  an  electric  spark  as  in  Lenoir's  engine,  or  by 
a  gas  Jet  as  in  Langen's  and  Hugon's.  The  calculation  of  the 
delivery  of  these  engines  is  therefore  similar  to  that  for  hot- 
air  engines.  Before  we  pass  on  to  it  we  shall  give  a  few  gen- 
eral considerations  which  find  their  application  in  all  gas  en- 
gines, and  which  we  borrow  in  large  part  from  the  work  of 
Prof.  Grashof — "Mesultate  der  Meclianisclien  Wdrmetheorie," — 
u]3on  which  we  shall  base  our  future  calculations. 

The  degree  of  heating  of  the  air  in  a  gas  engine  depends 
evidently  upon  the  ratio  of  illuminating  gas  and  atmospheric 
air  in  the  mixture,  as  also  upon  the  composition  of  the  gas. 
The  illuminating  gas  is  a  mixture  of  several  others,  more  espe- 
cially the  following :  1,  bicarbureted  hydrogen  or  defiant  gas ; 
2,  carbureted  hydrogen  or  fire-damp  ;  3,  hydrogen ;  4,  carbonic 
oxide ;  and  5,  nitrogen.  While  the  two  first  are  combinations  of 
carbon  with  hydrogen,  the  carbonic  oxide  consists  of  carbon 
and  oxygen.  During  complete  combustion,  the  carbon  forms 
with  the  oxygen  in  the  air  carbonic  acid,  while  the  hydrogen 
and  oxygen  unite  to  form  steam.  These  products  are  there- 
fore, after  the  combustion,  mixed  with  the  heated  air. 

One  cubic  meter  of  illuminating  gas  (especially  from  hard 
coal)  consists  on  the  average  of 

0.42  cubic  meters  of  carbureted  hydrogen  gas  (CH,), 

0.08        "  "        bicarbureted      "  "    (C2H4), 

0.40         "  "         hydrogen, 

0.07         "  "        carbonic  oxide,  and 

0.03         "  "        nitrogen. 

The  weight  of  this  cubic  meter  at  0.76  meters  of  barometer 
and  15^  C.  is 

So  =  0.535  kilograms, 


GAS  ENGINES— CHEMICAL  DATA.  283 

while  that  of  1  cubic  meter  of  air  under  the   same  circum- 
stances is 

So  =  1.225  kilograms. 

The  density  of  the  gas  is  therefore 

^    _  -*%   _  0.535  _  ^   .cynrr 

1  hilogram  of  gas  gives,  by  complete  combustion. 

El  =  10430  heat  units, 

and  1  cubic  meter  of  gas  therefore  gives 

K=  SoK,  =  0.535  X  10430  =  5580  heat  units. 

The  weight  of  air  required  for  the  complete  combustion  of 
1  kilogram  of  gas  is 

Li  =  14.5  kilograms, 

and  1  cubic  meter  of  gas  requires  for  complete  combustion  an 
air  volume, 

J-      LiSo      14.5  X  0.535      no     i-        i 

L  =  — ^— y  =  zi-7;^?r^ =  6.3  cubic  meters. 

So  1.225 

If  we  mix  1  cubic  meter  of  gas  with  a  cubic  meters  of  air, 
the  weight  of  1  cubic  meter  of  the  mixture  is  evidently 

1.225a  +  0.535  ,  ., 

s  =  — z. kilograms. 

a  +  1  ^ 

For  example,  for  a  =  10 

12.25  +  0.535     .-.ac-,--, 
s  = Yi =  1.162  kilograms. 

The  density  of  the  mixture  is  then 

,  _  g  _  1.225CT  +  0.535  _  1.225  (a  +  0.4367)  _a  +  0.4367 
So       1.225  (a  +  1)  ~       1.225  (a  +  1)       ~      a  +  1     * 


For  the  preceding  example 


d=  1^0:^  =  0.9488. 


284  THEBM0DYNAMIG8. 

After  complete  combustion  of  tlie  gas,  tlie  density  of  tlie  re- 
sulting mixture  of  carbonic  acid,  steam,  nitrogen,  and  air  at 
15°  0.  is 

n  -  ^  +  Q-^8 
a  +  0:83' 

Since  here  the  numerator  is  greater  and  the  denominator  less 
than  in  the  preceding  expression  for  d,  the  density  has  in- 
creased. 

For  example,  for  a  =  10  we  have 

10  +  0.48  _  10.48  _ 

while  before  combustion  it  was  0.9488. 
In  general,  then,  the  increase  of  density  is 


i>  _      {a  +  0.48)  ( 
d       (a  +  0.83)  {a 

'a  +  1)               a^  +  1.48a 

+  0.48 

+  0.4367)  ~  a^  +  1.2667a 

+  0.3625' 

For  a  =     8, 

10,        12,       14,        we 

)  have 

§=--. 

1.020,  1.017,  1.014. 

The  specific  heat  of  the  mixture  is — 

for  constant  volume 

0.1684a  +  0.286 

c^ 

a  +  0.48       ' 

for  constant  pressure 

Cr, 

0.2375a  +  0.343 

^  a  +  0.48 

Let  us  seek  now  how  many  heat  units  are  imparted  to  each 
kilogram  of  the  mixture  of  a  +  1  cubic  meters  when  the  gas  is 
kindled. 

From  experiments  we  can  conclude  that  during  combustion 
about  I  to  ^  of  the  heat  is  conducted  away  by  the  cooling 
water,  and  therefore  only  |  to  |  of  the  gas  is  used  for  the  heat- 
ing of  the  air. 

If  we  assume  generally  that  the  atli  portion  of  1  cubic  meter 


GAS  ENGINES— CHEMICAL  DATA.  285 

of  gas  furnislies  the  heat  required  for  the  increase  of  expansion, 
this  last  is 

ah  heat  units. 

This  is  distributed  over  (a  +  1)  cubic  meters  weighing  {a  +  l)s 
kilograms.     Hence  each  kilogram  contains 

heat  units. 


(a  +  1) 


If  now,  originally,  the  absolute  temperature  of  the  gas  mix- 
ture is  Tq,  and  immediately  after  firing  Ti,  we  have 

rp    _    r^  1  Oih 


c    {a  +  l)s  ' 


If  we  denote  further,  the  original  pressure  at  To°  by  po  (kilo- 
grams per  square  meter),  and  that  after  the  firing  by  jh,  we 
should  have,  if  there  were  no  increase  of  density, 

Po       To ' 

Since,  however,  the  mixture  is  increased  in  density,  in  the 
ratio  of  D  to  d,  we  have 


or  putting  B^  for  -^-, 

Co 


n_p^ 

d    po 

T, 

'■%' 

T, 

'--k 

^1 

EXAMPLE. 

According  to  the  experiments  of  Tresca,  there  were  mixed  in  a  Lenoir  gas 
engine,  on  the  average,  13  parts  of  air  with  1  part  of  illuminating  gas.  What 
is — 1,  the  specific  heat  c„  for  constant  volume  ;  2,  the  increase  of  temperature 
T^  —  Tq  ;  and  3,  what  is  the  expansive  force  of  the  mixture  after  firing  ? 


286  TEEBM0DYNAMIG8. 

We  have  in  our  formulae  a  =  13,  hence 

_  0.1684x13 +  0.286 

'"  -  13  +  0.48 ^•^^^^- 

If  we  take  «  =  f , 

rp       T  -f    -t    -— I  X  5580  _i9qi° 

1,-1, -t,       to-  (j_^g3g    1.325  X  13  +  0.535  "  ^^"^^  ' 

If  we  take  T^  =  273  +  15°  =  288%  we  have 

T,  =  1231  +  288  =  1519°. 


For  a  =  13 


_D  _      W  +  1.48  X  13  +  0.48      _ 
'  "  d  ~  13'^ +  1.2667  X  13  +  0.3625  ~ 


Hence  the  pressure  after  firing  is 

^'  ~  1016  ^  -333-  =  5.19  atmospheres, 

if  ^0  is  1  atmosphere. 

Delivery  and  Gas  Consumption  of  the  Lenoir  Engine. — The  gas 
beliind  tlie  piston  of  this  engine  drives  it  forward  after  firing, 
by  reason  of  its  increased  tension,  and  thus  performs  work.  If 
the  cylinder  were  not  surrounded  by  cold  water,  and  the  gas 
could  therefore  expand  adiabatically,  and  if  we  had  to  do  only 
with  atmospheric  air,  we  should  have 


Since,  however,  heat  is  abstracted  during  expansion,  and  since, 
also,  the  value  of  k  (1.41),  or  the  number  by  which  the  specific 
heat  of  air  at  constant  volume  must  be  multiplied  in  order  to 
obtain  that  at  constant  pressure,  is  also  different,  because  we 
have  to  do  with  a  mixture  of  air,  carbonic  acid,  steam,  etc.,  the 
law  of  change  of  p  and  v  cannot  be  represented  by  this  ex- 
pression.    It  can,  however,  be  given  by  the  general  expression 

^m^«  _  pm^n       (page  195). 

For  this  case  the  specific  heat,  under  the  assumption  that 
we  have  to  do  with  air,  is 

mk  —  n 


m  —  n 
where  therefore  Tt  =  1.41  and  c„  =  0.1684 


LENOIR  GAS  ENGINE— BELIVERY.  287 

Now  we  have  found  in  tlie  present  case,  for  c„  the  value 

0.1684a  +  0.286  ,,  ,„   ^,  .        .  moo^x         ^   .       .^ 
TTrro (*°^  a  =  13  this  gives  c  =  0.1836)  and  for  the 

•^    ,      ,  p  ,     ,  1        0.2375«  + 0.343 

specmc  heat  lor  constant  pressure  or  c„  we  have -pr^-p: — , 

^  ^  a  +  0.48 

hence 

y_cp_  0.2375a  +  0.343  _  ^  ,_ 

c„  ~  0.1684a  +  0.286  "  ■^•'^^^• 

The  specific  heat  for  the  law 

is  therefore,  in  our  case, 

m  -^  —n 

Cy  _  mCp  —  nc„ 

' c„  —  , 


m  —  n 


We  can  now  find  the  delivery  during  expansion  by  the  for- 
mula 


■••■['-(?■)"] 


if  Vi  is  the  volume  of  the  mixture  before,  and  V  that  after  the 
expansion,  or  at  the  end  of  the  stroke  of  the  piston. 

If,  now,  the  cross-section  of  the  cylinder  or  the  area  of  the 
piston  is  F  sq.  meters,  the  entire  stroke  of  the  piston  s,  and 
the  distance  passed  over  during  entrance  of  the  gas  eiS,  where 
therefore  ei  is  a  proper  fraction,  we  have 

V~  Fs  ~^' 

The  value  of  ei  we  call  the  "  degree  of  fill." 
If  we  insert  it  in  the  above  formula  we  have 


i:  = 


7t  —  m 


L  =  Fs\ra,.^-f] 


288  THERMODYNAMICS. 

The  work  of  the  back  pressure  of  tlie  air  is 
Fpo  {s  -  e^s)  =  Fpos  (1  -  e,). 
Hence  the  effective  delivery  per  stroke  is 


Fs 


['^'P^'t:^-^'^^^-'^^} 


If  there  are  u  revolutions  per  minute,  we  have  for  the  de- 
livery per  second 

Ls=^Fs  Imp,  ^-^^^'  -  i^o  (1  -  ei)1  meter-kils.  .  (LXXIII.) 

If  we  denote  the  efficiency  by  y,  we  have  for  the  actual  or 
useful  delivery  per  second 

Lu  =  y  ^Fs\  mpi ~Po(^  —  ^i)     meter-kilograms. 

If  we  put 

n 

we  have 

Lii  =  uFsl  meter-kilograms. 

If  the  horse-power  corresponding  is  N,  we  have 

N='^ (LXXV.) 

The  consumption  of  gas  per  revolution  is  found  from  the 
proportion 

2FeiS 


a  +  1 : 1  =  2FeiS  :x    or    x-- 

The  consumption  per  second  is  then 

uFeiS 
30(a  +  l)' 


a  +  1 


LENOIR  GAS  ENGINE— BE LIVEBY.  289 

Hence  the  gas  consumption  for  each  liorse-230wer  per  sec- 
ond is 

30(a  +  l)i\r' 

and  per  horse-power  per  hour 

V}S)uFe^s 

{a-\-\)N' 

Substituting  the  value  of  N 

G  =  ^^  ^  cubic  meters .     .     (LXXYI.) 

According  to  Grashof,  the  above  formulae  give  results  agree- 
ing well  with  experiment,  when  we  put  m  =  1  and  n  =  2.  Hence 
the  law  of  pressure  variation  is  given  by  the  simple  equation 


The  curve  representing  this  law  approaches  the  axis  of  ab- 
scissas more  rapidly  than  the  adiabatic  (page  194).  If  we  insert 
these  values  of  m  and  n  in  Equation  LXXIV.,  we  have 

or  putting  ^o  =  10334, 

I  =  344.4?/  (1  -  ei)  {^  ei  -  l)  .     (LXXVII.) 

If  we  determine,  according  to  this  formula,  the  value  of  I  for 
different  values  of  ei,  and  then  find  the  corresponding  values  of 
G,  it  will  appear  that  only  for  a  certain  value  of  e^  will  the 
economy,  or  the  ratio  of  the  delivery  to  the  consumption  of  gas, 
be  a  maximum. 

Thus  for  a  =  13,  we  have,  as  already  found, 


19 


^^-5.19. 


290  THERMODYNAMICS. 

The  efficiency  y  is  on  the  average  0.57.  If  we  take  it  at  0.55, 
we  have,  putting  for  e^  the  values 

ei^0.40  0.45  0.50  0.55  0.60, 

for? 
?  =  122.4         139.1  151.1         158.1  160.2, 

and  for  G 
G  =  2.12  2.08  2.13  2.24  2.41. 

Hence  we  see  that  the  economy  is  greatest  for  e-i  about  0.50. 
Under  the  above  assumptions,  then,  the  cylinder  must  be  half 
filled  with  the  mixture  before  it  is  fired. 

Example  1. — What  is  the  delivery  of  a  gas  engine,  the  diameter  of  cylinder 
being  3  decimeters,  and  stroke  4  decimeters,  when  e^  =  0.50,  and  the  number  u 
of  revolutions  per  minute  is  60,  the  efficiency  being  taken  at  0.25  ? 

The  cross-section  of  the  cylinder,  or  the  area  F  of  the  piston,  is 

F=  3.1416  X  (0.1)^  =  0.0314  sq.  meters. 

Since  in  the  present  case 

I  =  151.1, 

we  have  for  iV" 

^^      60  X  0.0314  X  0.4  x  151.1       ,  ^„  ^ 

iv  = ^ =  1.52  horse-power. 

The  gas  consumption  per  hour  would  therefore  be 

1.52  x  2.13  =  3.2376  cubic  meters. 


Example  2. — What  must  be  the  volume  of  the  cylinder  of  a  two-horse-power 
engine,  which  makes  70  revolutions  per  minute,  when  e  i ,  or  the  coefficient  of  fill, 
and  y,  or  the  efficiency,  ai'e  both  0.55  ? 

From  the  formula  iV  =  ,  we  have 

75 

Fs  =  — J-  .    In  the  present  ease,  I  =  158.1,  hence 

„  75  X  2  150        „^,^^      ^.         . 

'        ^'  =  70  X  158.1  =  11067  =  ^-^^^^  '"^''  ^'^'''• 

Quantity  of  Cooling  Water. — It  remains  now  to  calculate  the 
amount  of  heat  which  must  be  abstracted  from  the  cylinder 
for  every  cubic  meter  of  gas  used. 


LENOIB  GAS  ENGmE-COOLINa    WATER.  291 

We  assume  tliat  directly  upon  firing,  so  mucli  lieat  i-s  at  once 
abstracted  as  is  contained  by  1  —  ex  cubic  meters  of  gas.  This 
heat  is 

(1  —  a)  h  lie  at  units. 

If  we  put  a  — -  I,  we  have 

J-  X  5580  =  1860  heat  units. 

Heat  is  now  abstracted  from  the  gas  mixture  during  expan- 
sion, since  the  law  of  pressure  variation  is  given  by 

In  order  to  find  this  heat,  we  must  first  determine  the  tem- 
perature at  the  end  of  expansion  or  at  the  end  of  the  stroke. 
This,  as  well  as  the  final  pressure,  can  be  easily  found. 

Denote  the  last  by  ^j,  then,  according  to  our  law. 

If,  for  example,  e^  =  0.50,  then 

p  =  5.19  (0.50)'^  =  5.19  X  1  =  1.297  atmospheres. 

The  temperature  T  corresj^onding  to  this  pressure  is  found 
from 

n  —  ni 

Since  T^  =  1519  (page  286),  we  have 

r=1519  X  0.50  =  759.5°     or 
t  =  759.5  -  273  =  486.5°. 
If  we  take  Si  =  0.45,  we  have 

p  =  5.19  (0.45)2  =  5.19  X  0.2025  =  1.051  atmospheres, 
and  the  absolute  temperature  T  at  the  end  of  stroke  is 
r=  1519  X  0.45  =  683.6°     or 
t  =  683.6  -  273  =  410.6°. 


292  THERMODYNAMICS. 

We  see,  therefore,  that  even  at  the  end  of  expansion,  and  in 
spite  of  the  withdrawal  of  heat  by  the  cooling  water,  the  tem- 
perature of  the  gas  mixture  is  still  considerable,  and  hence  that 
in  course  of  time  the  engine  may  suffer  injury. 

The  heat  abstracted  during  expansion  is  found  as  follows : 
From  a  previous  formula,  the  heat  abstracted  for  1  kilogram 
of  air  is 

m  — w 

In  our  present  case,  where  k  =  1.386,  m  —1,  n  =  2,  and 
c  =  0.1836,  we  have 

Q  =  ^'^f^  ~  ^  X  0.1836  (Ti  -  T)=  0.1127  (Ti-  T). 
-L  —  A 

For  ei  =  0.50,  we  found  T  =  759.5,  hence 

Q  =  0.1127  (1519  -  760)  =  85.5  heat  units. 

The  gas  mixture  of  a  +  1  cubic  meter  weighs 

1.225«  +  0.535  kilograms, 

or  since  we  have  taken  a  =  13,  16.46  kilograms.  Hence  the 
heat  abstracted  during  expansion  from  each  cubic  meter  of 
mixture  is 

85.5  X  16.46  =  1407  heat  units. 

After  the  exhaust  valve  is  opened,  the  pressure  falls  rajDidly 
from  jj  to  that  of  the  atmosphere  po?  and  that  portion  of  the  heat 
disappears  which  went  to  increase  the  rectilinear  motion  of  the 
gas  molecules.  The  mixture  then  flows  out  under  the  constant 
pressure  jjq. 

If,  disregarding  this  loss  of  heat,  we  allow  the  gas  at  the 
absolute  temperature  Tc,  =  273  +  150  =  423°  to  issue  under 
constant  pressure  p^^,  we  must  abstract 

(1.225a  +  0.535)  Cr{T -  T^)  heat  units. 

Since  Ci  =  0.2545,  and  T  in  our  example  is  759.5  or  760,  we 
have 

16.46  X  0.2545  x  337  =  1414  heat  units. 


HUGON'8  ENGINE.  293 

Hence  tlie  entire  lieat  abstracted  is 

1860  +  1407  +  1414  =  4681  heat  units. 

Since  a  one-liorse-power  gas  engine,  with  coefficient  of  fill  of 
0.50,  requires  j)er  hour  2.13  cubic  meters  gas,  the  heat  ab- 
stracted is 

4681  X  2.13  =  9970  heat  units. 

If  the  cooling  water  is  heated  50^,  say  from  10^  to  60°,  we 
have  for  the  weight  of  water  W  which  must  circulate  per  hour 
around  the  cylinder, 

mW=miO     or     TF=  199.5  kilograms. 

This  weight  is,  however,  too  great,  as  we  disregard,  as  above 
remarked,  that  heat  which  disappears  suddenly  when  the  ex- 
haust valve  opens. 

Hugons  Engine. — In  this  the  heating  of  the  gas  mixture  is  less, 
by  reason  of  the  cold  water  injected  into  the  cylinder,  than  in 
the  Lenoir  engine,  and  the  piston  packing  suffers  less.  Ac- 
cording to  Grashof,  we  have  here  for  I  (page  288), 

I  =  344.4?/  [7.21  (ei  -  e,^-')  -  1  +  ej]     .     .     (LXXYIII.) 

where  y  can  be  taken  at  0.55. 

If  a  — 13,  and  the  weight  of  water  injected  per  cubic  meter 
of  gas  used  is  2  kilograms,  we   have,  according  to  the  same 

author,  —  --  4.33,  and  in  the  expression  which  gives  the  law  of 

variation  of  pressure,  we  have  %  =  1.6. 


For  e    =      0.4 

0.45 

0.5 

0.55 

/    =117.4 

129.8 

137.6 

141.2 

and  G  =     2.19 

2.23 

2.34 

2.50  cubic  meters. 

The  gas  jets  for  firing  require  about  0.25  cubic  meters  of  gas 
per  hour  additional. 

From  the  values  of  I  we  can  find  the  delivery  by  the  same 

formula  as  before, 

,,     uFsl  , 

JS  =  — r^K^  horse-power. 


294  THEBM0DTNAMIC8. 

Descrijjtion  of  the  Atmospheric  Gas  Engine. — Tliis  engine  is, 
beyond  doubt,  one  of  the  finest  inventions  of  tlie  century,  and 
was  justly  awarded  tlie  gold  medal  by  the  jury  at  the  Exposi- 
tion in  Paris,  1867.  In  fact,  the  inventor  has  overcome  in  a 
very  ingenious  manner  the  disadvantages  connected  with  the 
engines  of  Lenoir  and  Hugon,  and  the  essential  princijDles  of 
the  machine  are  based  upon  the  results  of  scientific  investiga- 
tion. The  construction  also,  in  the  opinion  of  the  author,  leaves 
little  to  be  desired. 

The  manner  in  which  the  inventor  has  happily  overcome  the 
defects  of  the  older  gas  engines  is  best  told  in  his  own  words 
{Dingier  s  Fohjt.  Journal,  Bd.  clxxxvi.). 

"By  the  combustion  of  explosive  mixtures  in  a  confined 
space,  the  products  of  combustion  are  heated  by  the  heat  dis- 
engaged, and  in  consequence  have  a  tendency  to  expand,  which, 
when  prevented,  gives  rise  to  a  pressure  upon  the  inclosing 
walls  corresponding  to  the  temperature.  This  pressure  ex- 
ists so  long  as  the  combustion  products  lose  no  heat. 

"  If  they  are  cooled,  they  contract  under  the  pressure  of  the 
atmosphere,  which  directly  or  indirectly  surrounds  them. 

"  If  it  is  wished  to  utilize  the  pressure  as  a  motive  force, 
the  question  arises,  what  time  elapses  between  the  heat- 
ing and  cooling,  between  the  expansion  and  contraction  of  the 
gases. 

"  This  time  is  known  to  be  very  short,  and  we  lose  a  portion 
of  the  heat  generated  by  combustion,  by  conduction  and  radia- 
tion through  the  cylinder  sides,  when  we  endeavor  to  utilize 
the  expansion  of  the  heated  gases,  if  we  do  not  allow  them  to 
expand  quickly  after  the  combustion.  This  loss  of  heat  corre- 
sponds, of  course,  to  the  loss  of  a  portion  of  motive  power. 

"  If,  we  conceive  an  engine  as  ordinarily  constructed,  that  is, 
with  piston  connected  by  connecting-rod  and  crank  with  a  fly- 
wheel, we  shall  have  opposed  to  the  explosion  in  the  cylinder, 
back  of  the  j^iston,  not  only  the  useful  work,  but  also  the  mass 
of  the  entire  system.  Such  an  engine  must  work  with  enor- 
mous velocity,  and  the  action  must  be  an  impulsive  one,  and 
since  the  moving  masses  can  never  take  an  acceleration  corre- 
sponding to  the  intensity  of  the  explosion,  the  heat  not  utilized 
must  heat  considerably  the  inclosing  walls  of  the  cylinder. 

"Guided  by  such  considerations,  we  have  adopted  in  the 


ATM08PHEBIG  GAS  ENGINE. 


29a 


296 


THEBMOD  YNAMIG8. 


construction  of  our  en- 
gine tlie  principle  that 
a  direct  utilization  of 
the  explosive  force 
must  be  discarded. 

"  On  the  contrary,  we 
utilize  the  heat  set  free 
by  the  explosion, by  op- 
posing to  the  expansion 
of  the  products  of  com- 
bustion but  very  slight 
resistance,  and  by  em- 
ploying, as  the  motive 
power,  the  contractile 
action.  Thus  the  gases, 
as  soon  as  they  have 
lost  their  heat,  and 
hence  their  tension,  are 
compressed  by  the 
pressure  of  the  atmos- 
phere to  that  volume 
which,  after  cooling, 
corresponds  to  their 
temperature  and  chem- 
ical constitution. 

"  "With  these  pre- 
liminary remarks  we 
can  now  explain  more 
in  detail  the  construc- 
tion and  action  of  the 
engine. 

"  We  shall  make  use 
for  this  purpose  of 
Figs.  45  to  54.  Fig.  45 
is  a  A^ertical  section,  and 
Fig.  46  gives  a  vertical 
projection  and  other 
details  of  construction. 

"A  is  a  cast-iron  cj\- 
]  inder,   with    two    air- 


ATMOSPHERIC  GAS  ENGINE. 


297 


tight  screw  covers  B  and  B^.  This  cylinder  is  surrounded  up 
to  about  ^d  of  its  height  with  a  double  wall,  the  space  between, 
as  well  as  between  B  and  Bi,  being  filled  with  cold  water,  which 
serves  to  cool  the  cylinder  walls.  This  water  is  brought  from 
a  reservoir,  not  shown  in  our  Figure,  through  the  pipe  r,  and 
the  warm  water  carried  off  through  'i\.  This  simple  circulation 
of  the  water  is  sufficient,  in  order  to  keep  the  cylinder  jacket 
always  at  a  low  temperature,  without  any  renewal  of  the  cooling 
water  being  necessary. 

"R  is  a  metal  piston,  which  can  move  air  tight  up  and  down 
in  the  cylinder.  At  every  firing  of  the  gas  mixture  this  is  shot 
up  the  cylinder.  Thus  the  mixture  expands  suddenly  and  cools 
very  quickly,  so  that  its  tension  falls  far  below  that  of  the  at- 
mosphere, and  the  piston  is  driven  down  by  the  outer  air  press- 
ure. This  last  motion  is  that  which  is  transferred  to  the  en- 
gine and  utilized.     The  transference  is  effected  as  follows  : 

"  The  piston  K  is  fastened  to  a  toothed  rod  Ki,  which  ends 
above  in  a  cross-head  T  (Figs.  45  and  46),  moving  up  and  down 
vertically  in  guides  F,  F,  only  partly  shown.  Thus  the  vertical 
motion  of  the  piston  is  insured. 

"  The  cylinder  plate  carries  two  pair  of  standards,  of  which 
only  one  pair  L^  L^  are  visible  in  our  Figure.  These  support  the 
shaft  JV,  which  carries  on  the  left  the  fly-wheel  B,  and  on  the 
right  the  belt  pulley  P.  At  the  middle 
of  the  shaft  the  disc  S  is  keyed,  a  cross- 
section  of  which  is  given  in  Fig.  47. 
Upon  the  prolonged  bars  of  this  disc  are 
two  loose  discs  Si  8i,  between  which  a  - 
crown-wheel  is  fastened  by  means  of  four 
bolts  Avhich  may  be  seen  in  Fig.  45.  The 
teeth  of  the  piston-rod  Ki  engage  with 
the  teeth  of  this  crown-wheel.  When  the 
piston  and  rod  move  upward,  the  crown- 
wheel and  both  discs  Si  and  aS'i  turn  from  left  to  right,  and  re- 
volve loosely  upon  the  boss  of  the  middle  disc  >S'.  When  the 
piston  and  rod  descend,  the  crown-wheel  is  made  by  a  special 
arrangement  to  grip  tightly  the  circumference  of  the  disc  S,  and 
thus  to  impart  motion  to  the  shaft.  The  construction  of  this 
special  arrangement  is  different  according  to  the  power  of  the 
eneine. 


298  THERMODYNAMICS. 

"In  engines  of  small  size  (i  horse-power)  the  inner  surface  of 
the  crown-wheel  carries  a  number  of  eccentric  surfaces  ah,  ah, 
etc.  (Fig.  48).  Between  these  and  the  circumference  of  the  disc 
there  are  the  same  number  of  rollers,  c,  c,  etc.  If  the  crown 
turns  in  the  direction  of  the  arrow,  as  it  does  when  the  piston 
descends,  those  surfaces  which  lie  nearest  the  center  m,  press 
down  the  rollers  upon  the  disc  and  compel  it  to  take  part  in 
the  motion.  If  the  crown  revolves  in  the  reverse  direction, 
there  is  no  such  pressure,  and  the  small  rollers  c,  c,  roll  upon 
the  circumference  of  the  disc. 

"In  larger  engines  the  construction  is  similar.     Here  the 
rollers,  however,  do  not  lie  directly  upon  the  circumference  of 
the  disc,  but  upon  the  upper  surface  of  four 
wedge-shaped    bodies,  a,  h,  c,  d  (Fig.   45). 
Here,  also,  the  portions  of  the  eccentric  sur- 
faces nearest  the  center,  during  motion  from 
right  to  left,  press  the  rollers  against  the 
wedges,  and  these  upon  the  circumference 
of  the  disc,  so  that  motion  is  transferred. 
"If  the  wedges  are  not  to  slide  upon  the 
^^^^^  perijDhery  of  the  disc  S,  the  angle  which  the 

eccentric  surfaces  of  the  crown-wheel  make 
with  the  circumference  of  the  disc  must  be  less  than  the  angle 
of  friction  of  the  metals. 

"  Upon  the  shaft,  W  there  is,  besides  the  toothed  wheel  Zq, 
also  another  one,  Z^.  This  engages  with  the  wheel  Z  (Fig.  45), 
which  is  keyed  to  a  second  shaft  to,  parallel  with  W.  Upon  lu 
is  the  ratchet  wheel  Si,  while  the  two  eccentrics  E  and  E^  form 
one  piece  and  revolve  loosely  upon  the  shaft.  The  catch  S-^ 
couples  or  disconnects  the  eccentrics  with  the  shaft  lo,  accord- 
ing as  it  engages  with  the  ratchet  wheel  8^  or  is  shoved  back 
by  the  bolt  Aj. 

"  As  the  piston  descends,  the  projection  N  upon  the  piston- 
rod  strikes  down  the  lever  Ai,  raises  the  left  side  of  the  cacch 
8-2,  which  then  engages  with  a  tooth  of  the  ratchet  wheel  8i,  and 
then  the  catch  and  both  eccentrics  E  and  Ei  j^artake  of  the 
motion  of  the  shaft  to.  If  now  the  number  of  strokes  of  the 
piston  is  equal  to  the  revolutions  of  the  shaft  or  fly-wheel, 
which,  as  we  shall  see,  is  by  no  means  necessary,  hi  is  raised 
during  a  revolution  and  the  catch  82  released. 


ATM08PHEBIC  GAS  ENGINE. 


299 


"  As  soon  as  the  catch  is  released,  which  must  always  occur 
when  the  piston  is  not  far  from  the  bottom  of  the  cylinder,  both 
eccentrics  come  to  rest,  and  the  slide  Ci, 
which  is  moved  by  the  eccentric  E,  has  its 
central  position.  In  this  position  the  port 
iji  (Fig.  49)  corresponds  with  y  in  the  cylin- 
der jacket  and  y^  in  the  cover  (72-  This  is 
pressed  b}^  springs  /,/',  upon  the  slide,  so 
that  air  cannot  get  between  the  surfaces. 

"  While  the  slide  C'l  remains  thus  a  short 
time  in  its  central  position,  the  piston  com- 
pletes the  lower  portion  of  its  stroke  and 
presses  the  products  of  combustion  out 
through  the  ports  and  valve  v  into  the  air. 
At  the  same  time  the  lever  7?i  is "  depressed,  the  catch  82  en- 
gages the  ratchet  wheel  Si,  and  both  eccentrics  partake  of  the 
motion  of  the  shaft.  One  of  these,  E,  which 
moves  the  slide,  moves  downward,  as  is  seen 
from  Fig.  45,  and  the  slide  moves  down.  The 
other.  El,  moves  up.  As  this  moves  the  lever 
li,  which  in  our  figure  is  behind  li-i,  and  the 
right  end  of  which  is  below  the  tappet  N,  the 
lever  and  piston  are  raised.  During  this  mo- 
tion of  the  piston  the  descending  slide  has 
closed  the  canal  y,  and  made  another  connec- 
tion, which  is  represented  in  Fig.  50.  Here 
the  canal  x  in  the  cylinder  side,  which  lies 
near  y  (Fig.  46),  communicates  with  the  chan- 
nel a  of  the  slide  d,  and  thus  by  m  and  n  with  the  outer  air 
and  a  gas  receiver.  Thus  as  the  piston  is  raised  a  mixture  of 
gas  and  air  enters  the  cylinder. 

"As,  however,  the  slide  Ci  moves  down,  the  canal  q  has  been 
in  communication  with  m  and  n,  and  is  therefore  also  filled 
with  a  mixture  of  gas  and  air.  When  the  piston  has  its  lowest 
position,  this  canal  communicates,  as  we  see  in  Fig.  50,  with  the 
opening  a  in  the  cylinder,  in  which  a  lamp  h  is  burning,  and 
thus  the  mixture  kindled  in  q.  Now  the  slide  moves  rapidly 
upward  and  takes  the  position  shown  in  Fig.  51.  The  kindled 
gas  in  q  is  still  burning,  and  so  soon  as  q  thus  communicates 
with  the  canal  x,  the  gas  mixture  under  the  piston  in  the  cylin- 


300 


THERMOD  TNAMICS. 


der  is  also  kindled.  By  the  expansive  force  developed  by  the 
explosion  the  piston  is  shot  up  with  great  velocity,  and  the  ec- 
centric E  raises  the  slide  somewhat  more,  and 
then  returns  it  to  its  central  position.  The 
lever  li^  and  detent  Si  then  release  the  eccentric, 
and  this  as  well  as  the  slide  remain  in  their 
positions  until  the  above  operation  is  repeated. 
"  Let  us  now  point  out  how  the  action  can  be 
regulated,  or  the  number  of  strokes  made  inde- 
pendent of  the  number  of  revolutions  of  the 
crank. 

"  In  the  gas  conductor  is  a  cock  by  which  the 
relative  proportion  of  gas  and   air  can  be  so 
^le-  51.  regulated  that  the  piston  is  shot  up  by  the  ex- 

plosion only  a  certain  distance.  "We  can  thus  alter  the  per- 
formance of  the  engine  at  pleasure.  Since,  however,  the  use- 
ful effect  of  the  engine  is  greatest  for  a  certain 
height  of  throw,  it  is  desirable  so  to  regulate  the 
force  exerted  that  the  throw  of  the  piston  may  be 
always  the  same,  and  hence  independent  of  the  de- 
livery of  the  engine  in  a  given  time. 

"  The  inventors  accomplish  this  by  making  the 
number  of  piston  strokes  independent  of  the  num- 
ber of  revolutions  of  the  axis,  regarded  as  constant. 
For  this  purpose  they  have  a  cock  D  (Fig,  45)  in 
the  end  of  the  exhaust  pipe,  that  is  the  pipe  through 
which  the  products  of  combustion  escape,  which  ^'^^'  ^'" 
allows  these  products  to  escape  more  or  less  rapidly.  By  this 
arrangement  we  can  cause  the  piston  to  descend  more  or  less 
rapidly.  Since  the  piston  is  pressed  down  by  the  difference  of 
pressure  of  the  atmosphere  and  inclosed  gas  mixture,  it  moves 
with  the  velocity  of  the  periphery  of  the  crown-wheel  so  long  as 
the  pressure  in  the  cylinder  is  less  than  one  atmosphere.  As 
soon  as  this  pressure  is  reached,  the  piston  sinks  by  virtue  of 
its  own  weight.  If  now  D  is  but  little  opened,  it  sinks  more 
slowly,  if  much  opened,  more  rapidly,  while  the  velocity  of  the 
periphery  of  the  crown-wheel  and  disc  is  nearly  constant.  If  D 
is  fully  opened,  the  engine  works  with  its  maximum  power. 
If  the  cock  is  closed  so  much  that  the  escape  of  the  products 
of  combustion  is  retarded,  h^  remains  longer  in  its  raised  j^osi- 


ATMOSPHERIC  GAS  ENGINE. 


301 


tion,  and  tlie  valve  works  more  slowly.  Tlie  j)osition  of  the 
cock  D  depends  therefore  upon  tlie  delivery  desired.  Hence, 
under  irregular  resistance,  the  revo- 
lutions of  the  fly-wheel  can  be  regu- 
lated by  this  cock. 

"In  Figs.  52  to  54,  we  have  repre- 
sented the  slide  and  slide  surfaces  on 
the  cylinder  and  cover. 

"  If  we  wish  an  automatic  regulation, 
it  is  only  necessary  to  fit  the  cock  D 
with  a  governor  run  by  the  shaft  in  the 
usual  manner." 

From  this  description  it  will  not  be 
disputed  that  the  construction  of  the  atmospheric  gas  engine 
is  very  complete,  and  that  it  deserves  to  be  placed  among  the 
most  ingenious  inventions  of  the  century,  a 
century  which  includes  also  the  engine  of 
Ericsson. 

The  inventors  add  to  their  description  in 
the  Journal  already  cited  the  following : 

"  The  atmosj)heric  gas  engine  differs  essen- 
tially from  earlier  gas  engines  in  the  follow- 
ing points : 

"  1.  Regard  is  had  to  the  physical  principles 
noticed  in  the  introductory  remarks, 
"'"" ""  "  2.  The  action  of  the  piston  is  intermittent. 

"3.  A  special  construction  transfers  the  downward  motion  of 
the  piston  to  the  fly-wheel  shaft. 

"  4.  The  construction  of  the  valve  motion  and  the  slide  is  es- 
sentially different  from  other  gas  engines. 

"  5.  By  changing  the  number  of  strokes  of  piston  for  a  con- 
stant number  of  revolutions  of  the  shaft,  the  performance  can 
be  regulated." 

That  which  distinguishes  in  other  respects  this  gas  engine 
from  ordinary  hot-air  engines  is,  that  it  is  very  easily  set  in 
action,  and  requires  very  little  attendance.  It  is  equally  safe 
with  all  hot-air  engines. 

It  has  the  disadvantage,  as  compared  with  hot-air  engines, 
that  it  can  only  be  set  up  in  places  where  illuminating  gas  can 
be  obtained  or  manufactured  without  great  cost. 


Ip    "-    o) 

W   l| 

r         f 


302  TEEBMODTNAMIGS. 

Tiieory  of  the  Atmospheric  Gas  Engine. — Let  Tq  be  tlie  abso- 
lute temperature  of  a  gas  mixture,  consisting  of  a  cubic  meters 
of  air  and  1  cubic  meter  of  gas,  and  2\  tlie  absolute  tempera- 
ture after  firing,  then  the  increase  of  temperature  is,  according 
to  page  285,  where  it  is  assumed  that  of  the  heat  which  every 
cubic  meter  of  gas  furnishes  by  its  combustion,  1  —  a  heat 
units  are  directly  withdrawn, 

rp  rp   J-       ^^ 

'        '~c    1.225a  +  0.535 ' 

0.1684a  +  0.286 

where  c  = t^-t^ . 

a  +  0.48 

According  to  experiment,  the  ratio  of  air  to  gas  is  8  to  1.  If, 
then,  we  take  a  —  8,  we  have 

0.1684x8  +  0.286     ^^^^ 
c= g^^g =  0.190. 

If  we  put  now  a  =  |,  we  have,  since  h  —  5580, 

J^    f  X  5580  _  3720 
'        ""0.19       10.335    ~  1.964' 


T,-T,  =  1894°. 
For  To  =  273  +  15  =  288°,  we  have 

T,  =  1894  +  288  ==  2182°. 
The  increase  of  density  Bi  is,  from  page  284, 

_      a^  +  1.48ft  +  0.48      _  ^  .^ . 
'  ~  a'  -1-  1.2667ft  +  0.3625 

Hence  the  tension  after  the  firing  is 

1   ^1 


ATMOSPHERIC  GAS  ENGINE— THEORY.  303 

Inserting  numerical  values 

1      2182  1 

^^^  "  LOM  T88"  ^  LOM  ^  ^'^^  "  "^'^  atmospheres  (nearly). 

The  law  of  variation  during  rise  of  piston  may  be  repre- 
sented by  the  equation 

(In  the  Lenoir  engine  we  had  ?i  =  2.) 

The  work  which  a  gas  volume  performs  by  its  expansion 
when  the  law  of  variation  of  jDressure  with  volume  is 

is,  from  preceding  principles, 

11  —  m  —i 

Since  we  have  assumed  m  —  1,  in  the  present  case 

where  Y^  is  the  least  and   Y  the  greatest  volume,  or  Yx  the 
volume  before  and  T^that  after  expansion. 

If  we  denote  now  the  cross-section  of  the  cylinder  by  F 
square  meters,  the  height  to  which  piston  is  shot  up  measured 
from  bottom  of  cylinder  by  s  meters,  the  height  which  the  gas 
mixture  occupies  before  it  is  fired  by  e^s  meters,  where  e,  is  a 
proper  fraction,  then 


Fi  _  Fe^s  _ 


Y-  Fs  -^' 


and  we  have  again  as  before, 


L  =  ^Fse,p,{l-e,--'). 


304  THERMODYNAMICS. 

Inserting  n  —  1,  e^  and  2h  in  the  parenthesis,  and  denoting  the 
atmospheric  pressure  by  Pq,  we  have 


^^^'»[t,T^(l-^'"-)]- 


This  delivery  goes  in  several  directions. 

A  portion  of  it  imparts  living  force  to  the  piston,  and  shoots 
it  up  to  a  certain  height.  Another  goes  to  overcome  the  resist- 
ance of  the  air,  and  a  third  the  piston  friction. 

The  overcoming  of  the  resistance  of  the  air  requires  the  work_ 

Fsp,  (1  -  eO- 

If  we  denote  the  piston  friction  by  R,  we  have  the  work  for 
overcoming  it 

Rs{l-e,l 

or  if,  with  Grashof,  we  put  R  =  pFpQ, 

pFp^s  (1  -  ei). 

If  further,  P  is  the  weight  of  the  piston  and  rod,  the  work 
required  to  raise  it  to  the  height  s  —  CjS  =  s  (1  —  ei)  is 

Ps{l-e,). 

If  here  we  put  P  ~  (pFp^^,  we  have 

(pFp^s  (1  -  ei). 

These  three  works  together  must  equal  that  of  the  gas  during 
its  expansion.     Hence 

•  ~      Fspo\^-^{l-e^^-')] 
Lpo  11  —  1^  J 

=  Fsp)o  [1  -  ei  +  P  (1  -  ei)  +  'T'  (1  -  ei)], 
or 

^^(l-er-')  =  (l  +  .  +  ^)(l-.,).         - 


ATM08PHEBIG  GAS  ENGINE— THEORY.  305 

Putting  for  tlie  sake  of  brevity  ^  —  m, 

l-ei"-^_(l  +  p+cp)(l-ei) 


n  —  1  mci 

Since  now  p,  cp,  e,  and  m  may  be  regarded  as  known,  we  can 
determine  n  from  this  equation. 

According  to  experiments  by  Meidinger,  in  an  engine  of  the 
kind  in  question  of  ^  liorse-power,  i^  =  0.01767  sq.  meters, 
s  =  0.99  meters  (full  rise  of  piston),  e^  =  0.114  meters,  P  =  21.8 
kilograms,  and  B=7  kilograms. 

Hence 

P-'PFpo^  21.8,    or    ^  =  0.01767;  10334  =  ''-"^- 
Further 

B^pFpo  =  7,        or      P  =  0.01767 1  10334  ^Q-^^^- 

Substituting  these  values  in  our  last  equation, 

(1  +  0.038  +  0.119)  (1  -  0.114)  _  1  -  e,--' 
7.4  X  0.114  71-1 


1.215  =  ^^ — -^—-    or    1.215^1-2.215 

n  —  1 


ei"-i  + 1.215^  =  2.215. 
This  equation  gives  n  =  1.60  very  nearly,  thus 
(0.114)0.60  +1.115  X  1.60  =  2.2157, 


°^'  °^^^  10000  *°°  ^^^^** 
Hence  the  equation 


^1.60  rr^^-y^l-' 


gives  the  law  of  variation  of  pressure  with  volume  when  the 
piston  rises. 
As  soon  as  this  law  is  known  we  are  able  to  calculate  the 
20 


306  THERMODYNAMICS. 

pressure  and  temperature  at  tlie  end  of  tiie  jpiston  rise.     If  we 
denote  this  pressure  by^,  we  liave 


Since  p^  =  7.4    and    —  =  0.114,   we  have 

p  =  7.4 (0.114)i«''  =  7.4  X  0.03098  =  0.2293  atmosphere. 

That  is,  the  final  pressure  is  far  heloio  the  atmosp>heric. 

For  the  sake  of  clearness  we  have  represented  graphically  in 
Fig.  55  the  law  of  change  of  the  gas  during  the  piston  rise.     Oe 
is  the  distance  of  the  piston  from  the  bottom  of  the  cylinder  at 
the  moment  the  gas  is  fired.     0/  is  about  three,  Og  six  times 
^^  this    distance,    and    Oh    is 

I  the   entire    rise.     At  /  the 

I  pressure   is   about  1.35   at- 

mospheres. This  is  repre- 
sented by  the  line/6.  At  g 
the  pressure  is  gc  =  0.45. 
Joining  the  points  ahcd  we 
obtain  a  curve  which  gives 
the  change  of  pressure  with 
volume.  We  see  that  at  a 
distance  of  hardly  0.33  me- 
ters above  the  cylinder  bot- 
tom the  pressure  has  al- 
ready sunk  to  one  atmosphere.  The  line  em  =  hn  represents 
the  pressure  of  the  atmosphere  increased  by  the  weight  of  the 
piston.  From  e  to  k,  the  area  ahim  denotes  the  excess  of  work 
of  the  gas  above  that  required  for  overcoming  the  air  pressure, 
the  piston  friction,  and  for  raising  the  piston.  This  excess  im- 
parts to  the  piston  its  living  force,  by  virtue  of  which  it  con- 
tinues to  rise  from  h  to  h,  or  from  i  to  n,  overcoming  the  resist- 
ance of  the  air  and  piston  friction.  During  this  the  inclosed 
gas  furnishes  indeed  a  work  represented  by  icdhh.  If  therefore 
we  add  this  work  to  the  living  force  of  the  piston,  we  have  the 
work  which  goes  to  overcome  the  air  resistance,  the  piston 
friction,  and  to  raise  the  piston. 


ATMOSPHEBIG  GAS  ENGINE— THEORY.  307 

If  now  T  is  the  absolute  temperature  at  tlie  higliest  position 
of  tlie  piston,  we  liave 

rp  1.60  —  1 

^  =  (0.114)~^r-=,  (0.114)«-««  =  0.2717. 

Since  T,  =  2182°, 

2^=2182  X  0.2717  =  593°, 
or 

i  =  593  -  273  =  320°, 

a  temperature  still  pretty  higli. 

The  useful  work  which  the  engine  furnishes  is  now  per- 
formed during  the  descent  of  the  piston,  by  its  weight  and  by 
the  air  pressure.  This  work  is  not  entirely  applied  to  moving 
the  engine  however.  A  part  serves  to  compress  the  inclosed  air 
again  to  the  pressure  of  the  atmosphere,  and  then  to  drive  it 
out  of  the  exhaust  valve.  Since  this  compression  takes  place 
with  the  velocity  of  the  periphery  of  the  crown-wheel,  and  hence 
relatively  rather  slowly,  a  part  of  the  heat  set  free  has  time  to 
radiate  from  the  cylinder  walls,  and  another  part  is  absorbed 
by  the  cold  water  which  surrounds  the  cylinder  up  to  about  ^d 
of  its  height.  In  consequence  of  this  the  temperature  sinks 
during  the  compression,  and  the  departing  products  of  combus- 
tion have,  according  to  Meidinger,  only  about  200°  tempera- 
ture, or  an  absolute  temperature  of  200  +  273  =  473°.  Another 
part  of  the  work  goes  to  overcome  the  piston  friction. 

"We  can  now  easily  calculate  at  what  height  of  piston  above 
the  cylinder  bottom  the  exhaust  valve  is  opened,  or  at  what 
height  the  products  of  combustion  are  compressed  back  to  one 
of  atmosphere. 

From  known  principles  the  weight  (r  of  a  volume  of  air  V, 
pressure  p  and  absolute  temperature  T,  is 

RT 

In  our  engine,  where  the  air  is  not  pure,  but  a  mixture  of  air. 


308  TEERMODTNAMIGS. 

carbonic  acid,  etc.,  B  lias  evidently  a  different  value.    If  we  put 
for  it  Eu  the  weight  G  in  tlie  moment  after  firing  is 

Assuming  that  there  is  no  loss  of  air,  this  weight  must  remain 
the  same  at  every  position  of  the  piston,  and  therefore  when 
the  mixture  is  compressed  to  the  atmospheric  pressure  ^^o-  Ifj 
then,  the  absolute  temperature  is  T.^  and  the  gas  volume  is  Fe-^s, 
e^s  being  the  distance  of  the  piston  from  the  cylinder  bottom, 
we  have 

From  these  two  equations  we  obtain 

Po      e,T,      """^    ^      Po    T,'     ' 

Since  now,  p^,  e^,  T^,  and  2\  are  known,  we  can  calculate  eg. 
Insertina;  numerical  values,  v/e  have 


.  ,  0.114x473      .^Q5 


As  &  —  0.99™-,  the  distance  e^s  is 

0183  X  0.99  =  0.181  meters,  or  18.1  centimetem 
Let  the  law  of  change  during  the  descent  of  the  piston  be 

pv''^  ^p^v^\ 

p  and  ?;  being  specific  pressure  and  volume  at  the  highest 
position,  and  PqVq  at  the  height  e^s  of  piston. 
Then 


f AW -&)"-"■ 


ATM08PHEBIC  0A8  ENGINE— TEEOBT.  309 

For  the  rise  of  tlie  piston  we  liad 


i-m-m-- 


If  we  divide  the  two  equations  for  —  and  —  one  by  the  other, 
^  Po        Pi  -^ 

we  have 


Po 


From  this  we  can  calculate  ?2i. 
Inserting  numerical  values 


Hence 


(0.183)"' =  7.4  X  0.114'-'", 
Qh.  log  0.183  =  log  7.4  +  160  log  0.114. 


log  7.4  X  1.60  log  0.114  „  „  „. 

fh=~^ aTqq  or  %  =  0.866. 

loa:  0.183 


Therefore  the  law  of  relation  between  pressure  and  volume 
during  the  descent  of  the  piston  is 

pv''-'''=PoVo'''''. 

Now  we  can  calculate  the   mechanical  work  necessary  for 
compressing  the  products  of  combustion  from  p  to  po.     This  is 

Z  =  ^^-— y  ^o^egs  (1  -  62"' -  ^) 


^'^'  ""  0.861-1  "  ^'P'  ^0:134  -  ^'-^"DlST  • 


310  THERMODYNAMICS. 

Further,  the  air  pressure  and  descending  weight  have  to 
overcome  the  piston  friction.     This  is 

pi^Spo  (1  -  62). 

Hence  the  work  required  by  both  resistances  is 

The  work   furnished  by  the   air  pressure   and  descending 
weight  is,  however, 

Fsp,  [1  -  62  +  <?>  (1  -  62)]  =  Fsp,  (1  +  ^)  (1  -  62). 

If  we  subtract  the  preceding,  we  have  the  theoretical  effect- 
ive delivery  per  revolution, 

Fsp,[{l  +  cp){l-e,)-p{l-^^-'^^ 

=  Fspo[a  -\-9-P)  (1  -  e,)  -^1=^]  . 

If  there  are  u  revolutions  per  minute,  the  delivery  per  sec- 
ond is 

Ls=  ^  Fspo  I  (1  +  ^  -  p)  (1  —  62)  -  n  134  J  meter-kilograms. 

Or,  since  po  =  10334, 

A  =  m.2dFsu[{l  +  cp-p){l-e,)-  ^^]  .  (LXXX.) 

In  horse-power 
N=  2.296FSU  \jl  +  cp  -  p)  (1  -  e,)  -  ^~j^]  •     (LXXXI) 


ATMOSPHEBIG  GAS  ENGINE- THEORY.  311 


What  is  the  theoretical  delirery  of  the  atmospheric  gas  engine  experimented 
upon  by  Professor  Meidinger,  for  which  F  =  0.01767  square  meters,  s  =  O.OO""-, 
e,  =  0.114,  P  =  21.8  kilograms,  B  =  7  kilograms,  u  =  34,  and  T2  =  273  +  200 
=  473=  ? 

First,  according  to  the  preceding  calculations,  1  +  (p  —  p=  1 
+  0.119  -  0.038  =  1.081.  Further,  1  -  e^  =  1  -  0.183  =  0.817. 
Hence  {1  +  (p  -  p)  {1  -  e^)  =  1.081  x  0.817  =  0.8832. 

Then    e/-  =  0.2297   and    e^""^  -  e.  =  0.2297  -  0.183  =  0.0467. 

Therefore  ^yj^T  "  T^  "  ^'^^'^'     ^^^^^ 


(1+  cp-p)  {!-€,)  +  ^J^  =--  0.8832  -  0.349  =  0.5342. 

U.ld4: 

For  I^su  we  have 

•     0.01767  X  0.99  x  34  =  0.5947,     therefore 

jp'su  [(1  +  cp  -p){l  -  62)  -  -^^]  =  0.5947  x  0.5342  =  0.3177. 

If  this  result  is  finally  multiplied  by  172.23,  we  have  for  the 
theoretical  delivery  in  meter-kilograms  54.75. 

Experiment  gave  40  meter-kilograms,  and  therefore  the  effi- 
ciency is 

40 


54.75 


=  0.73. 


According  to  Grashof,  the  efficiency  of  the  engine  in  question, 
when  oiled  carefully  and  at  short  intervals,  is  given  by  the 
equation 

2/ =  0.838-0.054-, 


in  which  z  is  the  number  of  revolutions  of  the  fly-wheel  or  gear 


312  THERMODYNAMICS. 

shaft.     In  Meidinger's  experiment  z  was  75,  hence 

^=-0.838-0.54^  =  0.72, 

or  agreeing  almost  exactly  with  our  calculated  result. 
For  ordinary  practical  working  condition  Grashof  gives 

y  =  0.79-0.07-. 

Hence  the  actual  effect  of  an  atmospheric  gas  engine  is 

N^  2.mFsuy  [_{l  +  cp-  p)  (1  -  e,)  -  ^^~\  horse- 
power    .     .     .     (LXXXI.) 

From  a  circular  of  the  inventors  we  take  the  following : 

A  convenient  and  advantageous  motor  for  the  minor  indus- 
tries is  offered  by  the  Otto  Langen  gas  engine. 

This  engine  can  be  set  up  in  crowded  areas,  as  its  action  is 
entirely  without  danger  and  it  requires  little  space. 

The  consumption  of  illuminating  gas  per  hour  for  every  horse- 
power (actual)  depends  upon  the  size  of  the  engine,  and  is  on 
the  average  only  about  1  cubic  meter,  or  considerably  less 
when  the  engine  is  not  worked  up  to  its  limit. 

The  expense  for  gas  is  the  only  cost  of  working  ;  wages  for 
service  do  not  increase  with  size. 

The  water  for  cooling  requires  no  renewal.  Its  temperature 
does  not  exceed  50°.* 

*  According  to  Meidinger's  experiments,  the  cooling  water  for  tlie  engine  of  i  liorse-power  . 
was  70  liters.  The  circulation  was  maintained  independently,  and  during  10  hours  of  constant 
action  it  left  the  jacket  with  a  temperature  of  83°  C.  and  returned  with  67'^  C. 


CHAPTEE  XII. 

FOEMUL^    FOR    THE    VELOCITY    WITH    WHICH     AIR    FLOWS     OUT     OP 

VESSELS. 

In  Fig.  56  let  the  space  between  tlie  two  pistons  HI  and  G,  in 
tlie  vessel  ABCD,  and  the  pipe  EF,  be  filled  with  water  or  some 
other  liquid.  Let  the  piston  HI  be  pressed  toward  the  right 
with  p  kilograms  per 
square  meter,  and  G 
toward  the  left  with  p^ 
kilograms,  and  let  pi  < 
p.  If  the  area  of  the 
large  piston  is  F,  and 
that  of  the  smaller  /,  D  «  i 
the  total  pressure  upon  ^i*^-  56. 

one  is  Fp,  and  upon  the  other  fp^.  Let  the  distance  passed 
over  by  the  larger  piston  in  one  second  be  s,  and  that  passed 
over  by  the  other  be  s^.  Then  the  work  of  the  force  Fp  is  Fps, 
and  that  of  the  resistance /pi  is  ^i^i.  If  water  fills  the  space 
between  both  pistons,  then  for  every  position  Fs  =fsi,  or 
F:f=s,:s. 

"While  now  the  water  particles  pass  from  the  larger  to  the 
smaller  vessel,  they  must  take  a  greater  velocity ;  the  less  ve- 
locity s  passes  into  the  greater  s^.  If  one  cubic  unit  of  water 
weighs  y  kilograms,  we  have  Fsy  =fSiy.  The  work  Fps  of  the 
force  Fjo  has  not  only,  therefore,  to  overcome  the  work  fp^Si  of 
the  resistance  fpt,  but  also  has  to  give  to  the  weight  Fsy  = 
fsiy  a  greater  living  force.  "When,  then,  uniform  velocity  is 
attained  in  the  vessel  and  pipe,  we  have 


Fps  =^fp,s^  +  ^~^~-  Fsy, 


Fps  -fpiS^  = 


_«i  - 


2f/ 


Fsy. 


313 


314  JHERMODYNAMICS. 

That  is,  tlie  difference  of  tlie  work  of  the  force  and  of  the  re- 
sistance is  equal  to  the  increase  of  living  force  of  the  water, 
neglecting  the  loss  of  velocity  due  to  friction,  etc.  Such  an 
excess  of  work  must  always  exist  when  we  have  a  change  of 
velocity. 

If  s  is  very  small,  with  reference  to  Si,  we  may  neglect  5-^  and 
have  then 

Fps  -fpts,  =  ^  Fsy, 
or,  putting  lu  in  place  of  Sj, 

Fps-fpiS,  =  -^Fs (1). 


If,  now,  the  space  between  the  pistons  were  filled  with  an  ex- 
pansive fluid,  as  air,  instead  of  liquid,  the  case  would  be  some- 
what different.  Such  a  gas  would  expand  when  the  pressure 
on  one  end  was  less  than  that  on  the  other.  This  would  be 
especially  the  case  for  those  particles  in  the  vicinity  of  F,  as 
shown  by  the  dotted  lines.  If  we  assume,  as  before,  that  the 
volume  Fs  passes  per  second  into  the  pipe  FF,  then,  if  its 
weight  is  one  kilogram,  its  specific  volume  is  v.  This  volume 
increases  in  passing  out  to  v^,  so  that  Vi  =/«!.  The  works  Fps 
smdfpiSi  are  then  equal  topy  andpi^i,  and  we  have 

pv-p,v,=^^-l, 

where  1  is  the  weight  of  one  kilogram. 

If  we  assume  that  the  air  in  the  pipe  has  the  same  temper- 
ature as  in  the  vessel,  then,  by  Mariotte's  law, 

pv  =  p^Vi, 

1u 

and  the  left  side  of  our  equation  would  be  zero,  hence  ;v-  or  w 

would  be  zero.  "We  have  to  seek  the  cause  of  the  change  of 
velocity  in  the  nature  of  the  gas  itself.  If  we  examine  more 
closely  we  shall  recognize  a  force  which  causes  this  change. 
Thus,  in  order  that  the  temperature  may  be  constant  in  the 
pipe  and  vessel,  heat  must  be  imparted  from  without,  and  just 


EFFLUX  OF  AIB.  315 

so  mucli  lieat  as  is  equivalent  to  the  work  of  expansion,  or  to 
the  increase  of  living  force  of  the  molecules.     This  heat  is 

Q  =  2.^mQ  ART  los^, 

^  Pi 

and  the  equivalent  outer  work  is 

A  ^  pi 

We  have,  then,  this  expression  in  the  place  oipv  —  p^Vi  in 
Equation  (2).     Thus 

'^  =  2.d026BTlog^, 
^9  ^  Pi 


=  4/ 4:.6052B Tg  log  ^. 

y  Px 


pi 

Putting  in  the  place  of  JR  and  g  their  values  {g  =  9.81  meters). 


w;  =  36.365 /|/^  log  ^  .     .     (LXXXII.) 

We  can  make  use  of  this  formula  in  every  case  when  the  ex- 
pansion is  very  small ;  when,  therefore,  p  is  but  little  more  than 
Pi.  For,  in  such  case,  the  heat  required  is  but  little,  and  we 
may  assume  that  it  is  supplied  by  the  outer  air.  According  to 
Weisbach  this  is  always  the  case  when  j?  —  pi  is  less  than  -^-jjp. 

If,  however,  no  heat  is  imparted  during  the  expansion  from 
V  to  Vi,  this  must  be  supplied  by  the  heat  of  the  air  itself,  or,  in 
other  words,  it  must  lose  heat  equivalent  to  the  work  done  in 
expanding.     This  heat  is 

where  T-^  is  the  absolute  temperature  of  the  air  in  the  pipe. 
The  equivalent  work  is 

A       A^  ^ 


316  TEEBM0DTNAMIC8. 

Now  in  tlie  present  case,  by  the  combined  law  of  Mariotte 
and  Gay-Lussac, 

pv  _  PiVi 


and  since  T^  <  T,  we  must  have  2hVx  <  pv.     By  the  expansion, 

therefore,  the  specific  volume  increases  less  rapidly  than  the 

pressure  diminishes.     The  expression  pv  —  p^v^  in  Equation  (2) 

has  then  a  positive  value,  or  work  must  be  performed  by  the 

piston  HI,  in  order  that  the  air  may  flow  out  with  the  velocity 

w.     Hence  the  entire  work  necessary  to  impart  the  living  force 

iu 

-K—  to  one  kilos-ram  of  air,  is 


A  number  of  other  cases  may  be  conceived.  Thus  we  may 
suppose  heat  abstracted  during  the  expansion,  according  to  the 
law  pv~^  =  piVi ~ ^  =  etc.  Such  cases  have  no  practical  in- 
terest. 

If  in  the  above  equation  we  put  BT  iu  place  of  pv,  and  RTi 
ioY  piV^,  we  have 


'^^=  BT-  BT,  +  ^{T  -  T,)=  (r  +  ^)  {T  -  T,). 


Since,  however,  B  —  — ^^ — -. — - 


i{9' 


=  (z-1+1)(^-^')  =  t(^-^J 


If  we  assume  T  as  known,  we  can  easily  find  T^.   Thus,  since 
the  expansion  is  adiabatic,  we  have 


EFFLUX  OF  AIE. 

Also, 

<f)'''' 

hence 

T,= 

^ifr- 

We  have 

therefore 

for  10 

317 


=  |/2,^r[i-(fy-] 


44.449  y  T  [l  -  {^^j       J .  (LXXXIII. 


These  formnlse  apply  to  the  case  where  air  flows  out  of  a 
vessel  into  the  atmosphere.  In  this  case  p^  =  1.  Instead  of 
assuming  the  piston  HI,  by  moving  toward  the  right,  to  pre- 
serve a  constant  pressure,  we  may  suppose  fresh  air  constantly 
forced  in. 

Experiment  shows  that  the  above  velocity  is  never  attained 
completely.  The  particles  are  hindered  by  friction  and  mutual 
impact.  These  disturbances  cause  a  loss  of  velocity  which 
reappears  as  heat.  Since,  however,  the  velocity  of  efflux  for 
moderate  pressure  is  very  great,  the  heat  thus  generated  by  loss 
of  velocity  is  imparted  almost  entirely  to  the  particles  rather 
than  to  the  walls  of  the  vessel,  and  the  temperature  T^  at  the 
plane  of  the  orifice,  or  where  the  pressure  is  constant,  is  some- 
what greater  than  given  by 


r.=  r(A)-. 


Experiment  has  also  shown  that  the  greatest  velocity  is  not  in 
the  plane  of  the  orifice,  but  some  distance  from  it,  within  the 
pipe.  The  stream  possesses,  then,  at  this  point,  a  somewhat 
smaller  cross-section  than  the  orifice.  The  phenomenon  is 
almost  exactly  the  same  as  for  the  efflux  of  water  or  similar 
liquids. 


318  THERMODYNAMICS. 

Tlie  number  by  wliicli  tlie  area  of  the  orifice  must  be  multi- 
plied in  order  to  give  the  area  of  cross-section  of  the  stream 
where  the  velocity  is  greatest,  is  called  the  coefficient  of  con- 
traction. It  depends  not  only  upon  the  excess  of  pressure 
jp  —  ^1  in  the  vessel,  but  also  upon  the  form  of  the  orifice,  as 
well  as  upon  its  position. 

According  to  Weisbach,  for  an  effective  pressure  p  —  pxoi  50 
to  850""™-,  for  circular  orifice  in  thin  plate,  of  from  10  to  24:""'"- 
diameter,  the  coefficient  of  contraction  is  cv  =  0.566  to  0.811, 
increasing  with  p  —  p^.  If,  for  example,  the  area  of  orifice  is  1 
square  centimeter,  the  cross-section  of  the  stream  at  the  place 
of  greatest  velocity  is 

0.566  to  0.811  square  centimeters. 

But  in  this  cross-section  the  velocity  is  not  iv,  as  just  found, 
but  is  somewhat  less.  We  have  therefore  to  multiply  iv  by  a 
proper  fraction,  in  order  to  obtain  the  actual  velocity,  and  this 
fraction  we  call  the  "  coefficient  of  velocity."  If  we  denote  it  by 
cp,  and  the  effective  velocity  by  iVg,  we  have 

zOg  =  q)w. 

If,  now,  the  cross-section  of  the  orifice  is  F',  that  of  the 
greatest  velocity  will  be  «'i^,  and  since  the  particles  pass  with 
the  velocity  iv^  =  cpiv,  the  discharge  per  second  is 

V=  iVgaF  =  cpivaF. 

And  putting  for  lo  its  calculated  value. 


V=  4AM9cpaF 


/[-(fT^I- 


The  product  of  the  coefficients  of  contraction  and  velocity 
(aq))  by  which  we  multiply  the  area  F  of  the  orifice  and  the 
theoretical  velocity,  in  order  to  find  the  actual  discharge,  is 
called  the  " coefficient  of  discharge"  and  is  denoted  by  //.  Ac- 
cordingly 


■j/[-(fr] 


V=  UMQmFa/    1  -  ^    ■       I    .     (LXXXIV.) 


EFFLUX  OF  AIB.  319 

If  tlie  mouthpiece  tlirougli  wliicli  tlie  air  flows  is  fitted  to  the 
shape  of  the  stream,  we  have  cv  =  1,  because  the  orifice  has  then 
the  same  cross-section  as  that  where  the  velocity  is  greatest. 
For  such  orifices,  shaped  somewhat  as  shown  in  Fig.  56,  Weis- 
bach  gives  for  an  effective  pressure p—  p^oi  180  to  850™™-,  for 
area  of  orifice  of  about  10™™- 

(p=  fx  =  0.981. 

(One  atmosphere  is  760™™) 

For  circular  orifices  in  a  thin  plate,  from  10  to  24"""-  diameter 
and  an  effective  pressure  j:>  —  7^1  of  from  50  to  850"""-,  and  tak- 
ing cp  =  0.98, 

M  =  0.556    to    0.795. 

For  short  cylindrical  pipes  of  the  same  diameter  as  the  ori- 
fice, for  the  same  limits  of  pressure, 

jA  =  cp  =  0.737    to    0.839. 


ExAJiPLE  1. — With  what  velocity  iv^  will  air  flow  out  of  a  receiver  into  the  air, 
when  the  pressure^  is  1.033  atmospheres,  the  absolute  temperature  T  273  +  10  = 
283%  and  the  coefficient  of  velocity  <^  =  0.90  ? 

Since  the  pressure  is  but  little  in  excess  of  the  atmosphere,  we  may  make  use 
of  the  formula  LXXXIL,  page  315.     We  have  then 

We  =  0.9  X  36.365   1/283  log  1.033 


:  0.9  X  36.365  1^283  x  0.0141 


=  0.9  X  36.365  Vd.mOS 
=  0.9  X  72.367,    or  finally 
We  =  65.132  meters. 

Example  2.— In  a  receiver  we  have  air  under  the  constant  pressure  7)  =  1.2 
atmospheres.  The  absolute  temperature  is  T—27S+  10  =  283°.  With  what 
velocity  will  the  air  issue  when  the  discharge  takes  place  through  an  orifice  of 
the  shape  of  the  contracted  stream? 

Here  p  =  1.2  and  p^  =  1,  hence 


(^)' 


320  THERMODYNAMICS. 

Hence 


w  =  44.449  i/283  (1  -  0.9485)  =r  44.449  '^^14.5745, 
w  =  169.443  meters. 
Since  <p  =  0.981,  the  actual,  velocity  is 

We  =  (piv  ^  0.981  X  169.443  =  166.224  meters. 
If,  now,  in  the  equation 

2ff-A   ^^       ■^'^' 


we  put  We  in  place  of  w,  the  absolute  temperature  Ti  in  the  plane  of  the  orifice  • 
will  be 


T,  =  T 


We-      A 


883-r-fyxO. 


2g     cJc  2  X  9.81 

=  283  -  13.98  =  269.02. 


Hence  the  temperature,  Centigrade,  is  <i  =  369.03  —  373  =  —  4.02". 
We  see,  therefore,  that  even  for  a  slight  excess  of  pressure  of  only  0.2  atmos- 
pheres, there  is  a  considerable  reduction  of  temperature. 

Without  loss  of  velocity,  the  temperature  T^  would  have  been 


r,  =  283  -  ^y^-t^fr  X  0.00993  =  283  - 14.53  =  268.47°, 
«  X  y.oi 


or  268.47  -  273  =  4.53'  C.     By  the  resistance,  therefore,  the  air  is  heated  4.53  - 
4.03  =  0.51°. 

Example  3.— H  the  same  receiver  is  required  to  furnish  100  cubic  meters  of 
air  per  minute,  what  must  be  the  cross-section  i^of  the  orifice  of  discharge  ? 

y 
We  have  V—iVeaF,    or    i^—  —  ,     or 

since  a-l,        r=  ^  =  1.666,        m)^  s=  166.324, 


■F  =  ^QQ  gg^  =  0.01003  square  meters  =  1.003  square  dec. 


EFFLUX  OF  AIB.  321 

Finally,  we  remark  tliat  when  air  flows  from  a  receiver  into 
the  atmosphere,  the  contracted  stream  expands  again  in  a  man- 
ner similar  to  that  in  which  it  contracted.  It  thus  gradually 
loses  its  velocity  and  the  particles  come  to  rest.     It  is  evident 

that  the  entire  living  force  ^  ,  inherent  in  one  kilogram,  is  thus 

transformed  into  heat,  and  that,  therefore,  as  soon  as  rest  ob- 
tains, the  temperature  of  the  air  is  again  the  original  temper- 
ature T  which  it  had  in  the  receiver  before  expanding. 
21 


CHAPTEK    XIII. 

AIE  COMPRESSORS  AND    COMPRESSED   AIR  ENGINES. 

[The  following  pages  comprise  an  abstract  of  a  work  entitled  "  L'Air  Comprime,"  by  M.  A. 
Pernolet,  Paris,  ISTG.  The  abstract  formed  portion  of  a  Graduation  Thesis  written  by  Mr.  Bailey 
Willis,  M.E.,  while  a  student  in  the  School  of  Mines,  Columbia  College,  and  is,  with  his  permis- 
sion, given  here  with  insignificant  changes,  precisely  as  prepared  by  him.  Mr.  Willis  has  in 
several  places  found  occasion  to  differ  from  M.  Pernolet,  and  such  differences  will  be  found 
noted  in  the  Text.  Mr.  Willis  has  also  converted  all  the  fornnilce  and  calculations  into  English 
measures.] 

Work  of  Compression. — Suppose  we  have  a  given  weight  G  of  air,  whose  vol- 
ume is  F,,  pressure ^i,  and  absolute  temperature  T, .  By  means  of  a  piston  let 
this  air  be  compressed  adiabaticaliy  to  the  volume  V2,  pressure^,,  and  tempera- 
ture T2.  During  compression,  the  pressure^,  of  outside  air  acts  upon  the  piston 
to  help  compression.  After  compression,  the  volume  Fj  is  forced  under  the 
pressure  ^3  into  a  reservoir.     Then  the  work  of  compression  is 


But 


and 


hence 


i'2  "^^2  =  2  (c,,  -  c,)  To 


_p,F,  =|(c,-c,.)r,  , 


i.=  |(c„  +  c,-c„)(T,-^0, 


Lc=^c,{T,-T,) 


100.704<y(r2-^,)  meter-kil.     \     '    (^^^^^-^ 

183.36^(2'., -T,)  foot-lbs.  J 

322 


AIM  COMPRESSORS-WORE  OF  COMPRESSION.  323 

where  —  =  773,  and  degrees  are  measured  by  Fahrenheit  scale.     For  convenience 
of  use  we  may  write  these  equations  in  the  following  form  : 

L,  ^  100.704(?!ri  (^  -  l\  meter-kil.  1 

(LXXXVl.) 


\  (|f  -  l)  foot-lbs.       J 


Tc,  «Q 

We  can  find  from  the  table,  page  171,  -^  for  any  given  ratio  of  — -  ,  and  then 

find  the  work,  without  first  finding  T^. 

We  may  also  write  the  expression  for  the  work  of  compression  in  the  form 


,„,,n[(|f)'-i] 


100.704^r,  I  (  ^^  )       -  1  I  meter-kil. 


183.36^r,  [(— )  "  -ijfoot- 


(LXXXVII.) 


Again,  if  the  volume  Vi  of  air  to  be  compressed  is  given,  instead  of  the  weight 
G,  then  since 


we  have 


and  hence 


p.V,^^^{c,,-c,)T,, 


Ap,Y, 


(Lxxxviir.) 


Any  of  these  formulas  may  be  used  in  finding  the  work  of  compression,  as  may 
be  most  convenient,  and  the  table  on  page  171  will  greatly  facilitate  computa- 
tions. 

Volume  of  the  Compressing  Cylinder. — Let  the  volume  of  the  compressing 

cylinder  be  V,  and  the  volume  of  air  compressed  per  stroke  be  V^.    If  the  engine 

makes  n  revolutions  per  second,  it  will  make  2n  strokes  per  second,  and  the  vol- 

V] 
ume  of  air  compressed  at  each  stroke  is  ^  . 


324  THERMODYNAMICS. 

The  volume  of  the  cylinder  must  be  equal  to  this, 


But 


p^V^  =  GRT^,      hence 


GRT^ 


(LXXXIX.) 


Final  Temperature. — In  the  formulae  for  the  work  of  compression  already  de- 
duced, the  final  temperature  T.^  of  the  air,  as  it  passes  out  of  the  compressing 
cylinder,  occurs.     In  the  following  table  the  final  temperatures  for  different 

values  of  —  are  given,  the  initial  temperature  tx   being  assumed  equal  to  68' 

Fahr.,  or  T^  =  459.4  +  68  n=  537.4  °. 

Final  Temperature.— T^  =  459.4  +  68  =  537.4°,  or  t^  =  68°  Fahr. 


Pi 

Final  abso- 
lute temuer- 
ature  T^. 

Final   tem- 
perature in 
degrees 
Falir.  t^. 

Final  abso- 
lute temper- 
ature T2- 

Final  tem- 
perature in 

3 

644.8 
•735.3 
788.35 
841.1 
886.7 
937.3 
963.9 

185.4 

265.9 

338.95 

381.7 

437.3 

467.9 

504.5 

9 

997.43 
1037.95 
1058.3 
1084.3 
1109.6 
1133.85 
1156.6 

538.03 

3 

10 

11 

568.55 

4 

598.8 

5 

12 

634.8 

6 

13 

650.3 

7.     . 

14     . 

674.45 

8      ... 

15 

697.3 

These  values  are  easily  calculated  from  the  table  of  ratios,  page  171,  and  the 
table  can  be  readily  extended  if  desired. 

They  can,  if  desired,  be  easily  reduced  to  Centigrade  degrees,  or  the  Centi- 
grade values  can  be  calculated  directly. 

Compression  in  two  Cylinders  ivitJi  Intermediate  Reservoir. — We  see  from  the 
expression  for  the  work  done  in  compression 

L.  =  -^c,{T,-T,) 


that  in  order  to  reduce  the  work  of  compression  for  any  given  ratio  —  ,  we  must, 

if  possible,  reduce  the  final  temperature  T.,. 

This  may  be  in  some  measure  accomplished  by  compressing  the  air  in  one 


AIR  COMPRESSORS—TWO  CYLINDERS. 


325 


cylinder  to  a  pressure  2^1'   intermediate  between  p^  and  p^,  then  cooling  it 
under  the  constant  pressure  p,^'  to  the  temperature  T^,  and  then  further  com- 
pressing it  to  the  pressure  ^j 3  in  a  second  cylinder. 
The  work  of  compression  in  the  first  cylinder  is  then 


A  A 


and  in  the  second  cylinder 


[(If)  '  -]• 


The  total  work  of  compression  is  therefore 


L:  =  ^c,.T, 


[(tV^C^)^-} 


(XC.) 


Now  Lc  is  a  minimum  when 


^-1  A-l 

'         +  M-",  IS  a  minimum. 

/'a 


In  any  given  case,  _Pi  and  ^2  will  be  known,  and  hence  the  above  expression 

is  of  the  form 

X      h 
-  +  -. 
a      X 


Differentiating  and  placing  the  first  differential  coefficient  equal  to  zero,  we 


find 


X  =  Vab, 


and  this  value  substituted  in  the  second  differential  coefficient  gives  a  positive 
result. 

Hence  the  work  of  compression  will  be  a  minimum  when 


Therefore 


p.'   =     Vp.Po, 

PJ^L-^ 

Pi    Pi 

-=^l'^-.  [(#'-]" 

.    .... 

...,...-..,  [(I.f'-.] 

(XCI.) 


(XCII.) 


326 

THERMO  DT. 

It  follows  also, 

from 

P2 
Pi 

_  P'- 
Pz 

that 

T." 

or 

that  is,  the  final  temperatures  in  the  two  cylinders  are  equal. 
Since 


2V 
T, 


2* 


.  (xciir.) 


and  since  for  a  single  cylinder  and  the  same  ratio 


we  have 


T,  -  \pj        ' 


(XCIV.) 


Comparing  the  work  of  compression  in  two  cylinders  with  that  in  one,  we 
have 


h^i{^'-^]  if) 


k-l 
2k 


.    .    (XCV. 


+  1 


m 


(XCVI.) 


+  1 


Now  —  is  always  greater  than  1,  therefore  [yjr)    +  1  is  always  greater  than 

2,  and  Lc,  the  work  of  compression  in  one  cylinder  for  the  same  limits  of  ^,  and 
po,  is  always  greater  than  the  work  of  compression  in  two  cylinders. 

In  the  following  table,  the  ratios  y-  and  the  final  temperatures  in  degrees 

Fahrenheit,  for  one  and  two  cylinders,  are  given  ;  it  being  assiimed  that  T^ 
=r  4591.4  +  68  =  527.4  or  t^  —  68°  Fahrenheit. 
We  find  T^'  from  XCIII. 

T2  may  be  found  by  aid  of  the  table,  page  171; 


AIR   COMPRESSOBS—TWO   CTLINDEES. 


327 


TABLE, 


SHOWING   FINAL   TEMPERATURE   FOR   ONE  AND   TWO   CYLINDERS  AND   RATIO 


1/  ' 


r=  459.4  +  ^. 


i'2 

Final  temperature 
in  degrees  Fahr. 

Lc- 

Ih' 

Final  temperature 
in  degrees  Fahr. 

U 

Ih' 

One 
cylinder 

Two 

cylinders 

f./. 

One 
cylinder 

Two 
cylinders 

n- 

2 

3 

4 

5 

6 

7 

8 

185.4 

265.9 

338.95 

381.7 

427.3 

467.9 

504.5 

123.8 
159.1 
185.4 
207.5 
224.5 
239.9 
253.6 

0.95 
0.92 
0.90 

0.88 
0.87 
0.86 
0.85 

9 

10 

11 

12 

13 

14 

15 

538.0 

568.55 

598.8 

624.8 

650.2 

674.45 

697.2 

265.9 

277.0 
287.3 
296.8 
805.6 
313.9 
321.6 

0.84 
0.83 
0.83 
0.82 
0.82 
0.81 
0.80 

Friction  of  Air  in  Pipes. — The  compressed  air  is  conveyed  by  means  of  pipes 
from  the  reservoir  to  the  point  at  which  it  is  desired  to  utilize  it,  and  therefore 
there  is  a  certain  loss  of  pressure  due  to  friction. 

This  loss  is  independent  of  any  changes  of  temperature  of  the  air.  It  is 
directly  proportional  to  the  length  of  the  pipe  and  the  square  of  the  velocity,  and 
inversely  as  the  diameter  of  the  pipe. 

Denoting  the  loss  of  pressure  due  to  friction  by  F,  we  have  from  experiments 
at  Mont  Cenis 


F=  0.00936 


(XCVII.) 


where  u  is  the  velocity  in  feet  per  second,  I  the  length  of  pipe  in  feet,  and  d  the 
diameter  of  the  pipe  in  inches. 

In  conveying  the  air  through  the  pipe,  there  is  also  a  loss  of  power  due  to  the 
change  of  temperature.  If  we  denote  by  T^  the  temperature  at  the  point  of 
application,  we  have  for  this  loss 


F' 


(T,  -  T,) 


(XCVIII.) 


hence  T2  should  be  as  low  and  T-^  as  high  as  possible. 


The  Compressed  Air  Engine. — The  compressed  air  arrives  at  the  end  of  the 
pipe  with  a  pressure  p.^  and  at  an  absolute  temperature  T3,  while  G  pounds  of 
it,  the  amount  used  per  second,  occupy  a  volume  V3. 


328  TEEBM0DTNAMIC8. 

It  is  then  admitted  into  the  cylinder  of  the  compressed  air  engine  and  does 
work,  its  condition  at  the  end  of  the  stroke  being ^4',  T^',  and  F4'.  The  exhaust 
then  opens  and  the  air  escapes  into  the  atmosphere,  its  state  changing  to ^4,  Ti, 
and  1^4. 

The  air  may  perform  work  in  the  cylinder  in  three  different  ways. 

1.  It  may  act  at  full  pressure  during  a  portion  of  the  stroke,  and  be  then  cut 
off  at  such  a  point  that  the  pressure  in  the  cylinder  falls  by  expansion  to  the 
pressure  of  the  atmosphere.     In  such  ease,  ^4',  T^,  and  V^  become  equal  to  p^, 

2.  The  air  may  act  at  full  pressure  during  the  whole  stroke.  In  this  case  ^4', 
Ti,  and  V^'  are  equal  to  ^.3,  T-^,  and  V^',  and  the  work  corresponding  to  the 
change  of  temperature  T-^  —  T^  is  lost. 

3.  The  air  may  act  at  full  pressure  during  a  portion  of  the  stroke,  and  be  then 
cut  off  at  such  a  point  that  the  pressure  within  the  cylinder  is  reduced  indeed  by 
expansion,  but  not  to  the  pressure  of  the  atmosphere.  In  this  case  ^3  is  greater 
thanjp4',  and ^4'  is  greater  thanp4. 

"We  may  call  these  three  cases  respectively  : 

1,  Complete  expansion ;    2,  Full  pressure ;  and  3,  Incomplete  expansion. 


1.  COMPLETE  EXPANSION. 

We  have  to  deal  in  this  case  with  the  initial  conditions  2>i,  V3,  and  T,^,  and 
the  final  conditions  p^,  V^,  and  T4  of  G  units  of  weight  of  air,  the  amount 
used  per  second. 


Final  Temperature. — The  expansion  takes  place  between  the  limits  p^  and 
Pi  according  to  the  adiabatic  law.     Hence 


2\_  fp,\-^_   /P,y 
The  ratio  -^  may  be  found  in  our  table,  page  m,  fof  the  givett  value  of  ^ . 


WorA;.— The  work  of  the  air  in  the  cylinder  is  made  tip  of  two 
parts,  the  work  p^  V^  at  full  pressure  plus  the  work  of  expansion.     The  dispos- 
able work  is  this  amount  less  the  work  of  the  back  pressure  ^4  F4. 
Hence 

L,^p,V,  +  ^cAT,-T,)-p,V,. 


C0MPBES8ED  AIB  ENOmE— COMPLETE  EXPANSION.     329 

This  reduces,  as  we  have  already  seen,  to 


Volume  of  the  Cylinder. — We  have  found  for  the  volume  of  the  compressing 
cylinder 

Zn  p-^^ 

In  an  entirely  analogous  manner  we  have  for  the  volume  of  the  cylinder  of 
the  air  engine 

G   T, 
V  =  R~  ^ (XCIX.) 

2n  p^  ^  ' 

where  n  is  the  number  of  revolutions  per  second. 

Weight  of  Air  per  Second. — Let  N  denote  the  number  of  horse-power  per 
second  required,  and  77  denote  the  efficiency  of  the  engine.     Then  since 

we  have  for  English  measures 

hence 

„  _  550^  N 

^-     7/c,    {T,-T,) 

3  N 


V    T,-T, 


per  second.   .     .     .    (C.) 


JEffitiency  of  the  Compressor  and  of  the  Engine.— het  Z  be  the  ratio  between 
the  work  of  the  compressor  Lc  and  the  work  of  the  air  engine  Ld.     We  have  then 


c=^  = 


{T,-T,)      T,(l-^\ 


g=     L     VW      J ^ci.) 


T, 


t-(tr] 


330 


THEBMOD  YNAMIGS. 


In  this  expression  the  ratio  of  the  terms  in  brackets  is  usually  nearly  equal  to  1. 

T 

It  is  therefore  to  the  ratio  ^  that  we  must  look  for  any  increase  in  the  effi- 

ciency  of  the  combination  of  the  compressor  and  the  air  engine. 

T,^  is  usually  the  temperature  of  the  atmosphere  at  the  air  engine,  and  we  see, 
therefore,  that  to  increase  the  efficiency  we  must  decrease  T^.  This  agrees  with 
what  has  been  said  in  discussing  the  compressor. 

A  rough  approximation  to  the  efficiency  in  any  given  case  may  be  arrived  at 
by  placing 

^-  To- 

The  values  thus  obtained  will  always  be  greater  than  the  true  values,  and  the 
latter  will  approach  the  nearer  to  it,  as  the  loss  by  friction  in  the  pipes  is  less. 

For  the  sake  of  future  comparison  merely,  we  give  in  the  following  table  the 
efficiency  for  complete  expansion  calculated  from  the  f ormida 


Ti  being  taken  equal  to  527.4°  Fahr. 


Pi 

C- 

^• 

2 

0.82 
0.73 
0.67 
0.63 
0.59 
0.57 
0.55 

9 

0.53 
0.51 
0.50 
0.49 
0.48 
0.47 
0.46 

3 

10 

4        ... 

11 

5 

12 

6 

13         .     . 

7 

14        

8 

15 

for 


These  values  can  be  taken  directly  from  the  table,  page  171,  in  the  column 


In  practice  these  efficiencies  are  reduced  to  less  than  half  of  these  theoretical 
values.  For  if  we  denote  the  work  of  the  engine  that  works  the  compressor  by 
L,  and  the  work  performed  by  the  air  engine  by  L,  and  assume  the  efficiencies 
of  the  compressor  and  air  engine  at  a  fair  average  at  0. 70,  we  have 

i<:=0.7i,     and    L' ^Q.lLa,    hence 


■U  Lie  Jbc 


COMPRESSED  AIR  ENGINE— COMPLETE  EXPANSION.      331 

Moreover,  it  must  be  remembered  that  for  long  distances  the  term 


^-1 

k 


fc-i 
k 
1 


has  a  very  considerable  influence,  and  reduces  the  value  of  C  very  materially. 

Construction. — We  may  easily  deduce  formulae  for  the  cut-ofE  which  will  pro- 
duce complete  expansion. 

The  volume  of  air  at  the  pressure^; 3  used  per  stroke  is  — ^.  When  this  ex- 
pands to  the  pressure  ^4,  the  volume  must  be  that  of  the  cylinder  V,  for  which 
we  have  already  deduced  a  value. 

We  have  by  the  adiabatic  law 

F 

^v.       .... 

V 
From  our  table,  page  171,  we  can  find  the  value  of  3«-  -^-,  for  any  given  value 

of  — ,  in  the  column  for  —  . 

Pi  v-z 

Let  now  S  be  the  stroke,  and  s  the  distance  traversed  by  the  piston  during 
admission  of  air.     Then 

Let  w  be  the  angular  velocity  of  the  crank,  and  t-^  the  duration  of  admission. 
Then 

S=  5-  (1  —  COS  oot{). 


1^ 
1  -  COS  oj^i  =  3  -^  r=  2  (  ^  '  * 

^       \pi 


[^-<m 


cot,  =  arc.  COS     1  -  2  ^  (OIL) 


If  the  time  of  entire  stroke  is  t,  we  have 

cot  =  7r=  3.14,     and  hence 
t^        ootj^        oot^ 

t    ~     Got    ~      7i 


(cm.) 


332 


THERMOD  YNAMIGS. 


In  the  last  column  of  the  table,  page  171,  vre  have  given  the  volumes  of 
calculated  for  the  corresponding  values  of  —  ,  in  the  first  column. 


Air  Engines  with  Two  Cylinders  and  Intermediate  Reservoir. — ^We  have 
already  given  the  formula  for  the  final  temperature  T^,  for  complete  expansion 
(page  328),  viz.  : 

Tc-l 

T, 


^3  \pj 


Assuming  T.^  =  527.4°,  which  corresponds  to  60°  Fahr.,  the  values  of  Ti,  given 
in  the  following  table,  may  easily  be  calculated  by  the  aid  of  table  on  page  171. 


TABLE 

FOR  FINAL  TEMPERATURE    T^. 

T-i  =  459.4  +  68  =  527.4. 


P3 
Ih' 

Final 

absolute 

temperature 

Final 
temperature 
Fahrenheit 

Pi 

Final 

absolute 

temperature 

'J\. 

Final 
temperature 
Fahrenheit 

2 

431.4 

383.5 

352.8 

330.7 

313.7 

299.98 

288.5 

-  28 

-  75.9 

-  106.6 

-  128.7 

-  145.7 

-  159.4 

-  170.9 

9 

278.9 
270.5 
263.1 
256.6 
250.7 
245.3 
240.5 

—  180.5 

3 

10 

—  188.9 

4 

11 

—  196  3 

5 

12 

—  202.9 

6 

13 

—  208.7 

7 

14            .    . 

—  214  1 

8 

15     

—  218.9 

In  discussing  the  air  compressor,  it  has  been  shown  that  the  final  temperature 
To  could  be  reduced  by  compressing  in  two  separate  cylinders  connected  by  a 
cooling  reservoir  (pages  324,  325).  In  a  similar  manner  the  very  low  temperature 
2^4,  given  in  the  preceding  table,  may  be  in  a  measure  avoided. 

The  air  expands  in  the  first  cylinder  to  a  pressure  p^'  which  is  greater  than 
P^.  It  then  passes  into  a  reservoir  and  is  heated  by  jets  of  hot  water,  under  the 
constant  pressure  ^^i',  to  its  original  temperature  T,.  It  then  passes  into  the 
second  cylinder,  and  is  there  allowed  to  expand  to  the  pressure  jj  4  of  the  atmos- 
phere. 

The  work  in  the  first  cylinder  is 


L^  =  ^c,  (T, 


'>-|-[-(f7)'} 


G03IPBE88ED  AIM  ENGINE— TWO   CYLINDEBS.  333 

The  work  in  the  second  cylinder  is 

..  =  |.<n-ro4o,r,[x-(|.f]. 
Hence  the  total  disposatle  -work  is 

--^'-t-(#-(frf]- 

As  already  shown  (page  325),  the  last  two  terms  are  a  minimum  when 
Pi  =  '\/2}-3pi  and  T^'  is  then  =  T^". 


Hence  we  have 


T'  O^ 

IjA  max.  ^  a  -T  C, 


Since 


we  have 


T,        \p,  J  -  \p. 


r,'_  (T,^^ 


(CV.) 


T,        \T, 

where  T^  denotes  the  final  temperature  for  one  cylinder.     Comparing  the  work 
in  one  cylinder  with  that  in  two,  we  have 

an  expression  entirely  similar  to  XCVI.,  but  in  this  ease  the  denominator  is  less 
than  2,  and  Lg  is  therefore  less  than  L,/. 

In  the  following  table  we  give  for  comparison  the  final  temperatures  t^  and  t^> 

in  Fahrenheit  degrees,  and  the  ratio  j!- ,  for  different  values  of  ^-^ ,  T-.^  being 
Ed  Pi 


334 


THERMOD  YNAMIC8. 


taken  =  527.4°,  or  ^3  =  68'  Fahr.     The  values  for  one  cylinder  are  taken  from 
the  table,  page  332,  and  for  two  can  be  calculated  from  CVI.* 


T.,  =:  459.4 


527.4\ 


i2i 

Final  temperature, 
Fahrenheit. 
One  cylinder. 

Final  temperature, 

Fahrenheit. 

Two  cylinder.?. 

2 

-  28 

-  75.9 

-  106.6 

-  128.7 

-  145.7 

-  159.4 

-  170.9 

-  180.5 

-  188.9 

-  196.3 

-  202.9 

-  208.7 

-  214.1 

-  218.9 

+     14.5 

-  10.2 

-  28.2 

-  41.8 

-  52.8 

-  61.7 

-  69.3 

-  75.8 

-  81.9 

-  86.9 

-  91.6 

-  95.7 

-  99.7 

-  103.3 

1.04 

1.08 

1.10 

1.12 

1.13 

1.14 

1.15 

1.16 

1.166 

1.172 

1.177 

1.183 

1.19 

1.194 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14            ... 

15 

FULL  PRESSURE. 


The  action  of  the  compressed  air  at  full  pressure  is  as  follows  :  The  air  enters 
the  cylinder  and  follows  the  piston  throughout  the  stroke  at  full  pressure  p^. 
The  pressure  and  volume  remain  the  same,  and  there  is  therefore  no  change  of 
temperature.  At  the  end  of  the  stroke,  the  exhaust  opens  and  the  pressure  of  the 
air  suddenly  falls  from  ^3  to  ^4,  with  a  corresponding  lowering  of  temperature 
from  T,  to  T,. 

Thus  the  work  is  performed  within  the  cylinder  without  any  change  of  the 
internal  energy  of  the  air,  and  the  work  corresponding  to  the  change  of  temper- 
ature from  T:i  to  T^  is  lost  in  the  exhaust. 

*  [Mr.  Willis  omits  at  this  point  a  discussion  by  Mens.  Pernolet,  from  which  he  deduces  an 
expression  for  the  ratio  -^  of  the  efficiencies  of  a  single  and  double  cylinder  air  engine  with 


the  compressor.      The  result  obtained  ' 


i'  . 


,  which,  Mr.  Willis  observes,  is  correct 


as  far  as  it  goes,  but  is  based  upon  the  supposition  that  ^  =  — ^ .    From  Eq.  CI.  we  see  that 

-'2 

T^  A;  - 1 


and  hence  the  value  deduced  for  ^    by  M.  Pernolet  is    approxi- 


mately true  for  short  distances  only.] 


COMPRESSED  AIR  ENGINE— FULL  PRESSURE 


335 


Disposable  Work. — The  available  or  disposable  work  for  one  second  is  equal 
to  the  pressure^j  less  the  back  pressure ^4,  into  the  volume  V.^  ;  F3  represent- 
ing the  volume  of  the  weight  G  of  air  used  per  second  at  the  pressure  p.^.   Hence 


But 


therefore 


La  =  V-APi  -p,). 


G  T, 


i.=  4fe.-o,)rYl-|i 


=  53.268 r,(?  (  1 


^\ (CVII.) 

in  English  measures. 

This  equation  gives  a  simple  value  for  the  disposable  work  in  terms  of  the 

initial  temperature  and  of  the  pressure  ratio  —  . 

But  for  the  purpose  of  further  discussion,  it  is  desirable  to  obtain  an  expression 
G 


Tj),  as  is  the  case  for  complete  expansion. 
This  may  be  readily  done  by  placing 


of  the  form  -~Cp{T 


T  ^/ 


c.) 


^'{^-f) 


c,{T,-T.), 


where  Tr  is  an  unknown  quantity,  and  (?;,  —  T,)  denotes  the  degree  of  expan- 
sion which  would  give,  with  complete  expansion,  the  same  disposable  work  as 
that  obtained  at  full  pressure. 

Solving  the  above  equation  we  obtain 


k      p3 


0.71  +  0.29 


(CVIII. 


From  this  formula  the  A^alues  in  the  following  table  have  been  calculated 


2     .      .. 

0.855 
0.807 
0.783 
0.768 
0.759 
0.752 
0.746 

9  

0.742 
0.739 
0.736 
0.734 
0.732 
0.731 
0.730 

3     

10  

4 

11  

5 

12 

6 

13 

14 

8 

15 

336  THERMODYNAMICS. 

It  is  important  to  distinguish  that  T^  and  T^,  the  temperature  of  exhaust  for 
full  pressure,  have  no  direct  relation  to  each  other. 
Thus 

Tx  is  a  theoretical  temperature,  which  has  no  actual  existence  in  the  action  of 
the  air  at  full  pressure.  It  simply  means,  that  if  an  engine  worked  with  com- 
plete expansion  between  T3  and  T^,  this  engine  would  do  the  same  work  as  the 
fuU-pressure  engine  does  between  T^  and  T^.* 


Final  Temperature. — The  final  temperature  T4  is  found  from 

T,  \p,y 


for  the  expansion,  on  opening  the  exhaust  takes  place  rapidly,  and  according  to 
the  adiabatie  law. 

This  relation  is  the  same  as  that  akeady  given  for  complete  expansion,  and 
the  values  of  T4,  in  table  on  page  332,  apply  also  for  full  pressure  for  the  same 

ratios  of  —  . 
P 

Comparison  of  Efficiencies  for  Full  Pressure  and  Complete  Expansion. — We 
have  already  found  an  expression  for  the  efllciency  of  an  air  engine  working  with 
complete  expansion,  and  its  compressor,  and  have  given  the  values  of  Cj  calcu- 
lated from  the  approximate  formula 

in  the  table,  page  330. 

If  we  denote  by  ?'  the  eflS.ciency  of  an  air  engine  acting  at  full  pressure,  with 
its  compressor,  we  have 


^  {c,  -  c,)  T 
r  — 


(-fr) 


-.[(^)""-0 


*  [5Ir.  Willis  calls  attention  to  this  point,  because  Mons.  Peniolet  distinctly  states  that  Tx  is 
T^,  and  tlien  deduces  the  quantities  given  in  his  tahle  viii.,  p.  62  of  '■^  L'air  comjirime,''''  under 
this  assumption.    The  results,  therefore,  of  the  last  three  columns  of  that  table  arc  incorrect.] 


C0MPBES8ED  AlB  ENGINE— FULL  PBE8SURE.  337 


Hence 


P^ 


'■    (ft) 


k       _1 


0.29 


1-  ^ 
Pi 


ify-^ 


(CIX.) 


It  being  assumed  that,  as  before, 


T,=T, 


11  =  1:1 
Pi      Pi 


T3  =  527.4°. 


Efficiency 

! 

Efficiency 

Jh 
2h' 

for 
complete  |       full 

^. 

for 
complete  ,       full 

^  ■ 

expansion 

pressure 

expansion;  pressure 

^• 

i'- 

i.     !     i'. 

2 

0.82 

0.65 

0.79 

9 

0.53          0.29 

0.55 

3 

0.73 

0.51 

0.70 

10. 

0.51     !     0.275 

0.54 

4 

0.67 

0.44 

0.66 

11 

0.50     ■     0.26 

0.52 

5 

0.63 

0.39 

0.62 

,  12 

0.49     i     0.25 

0.51 

6 

0.59 

0.35 

0.59 

13 

0.48         0.24 

0.50 

7 

0.57 

0.33 

0.58 

14 

0.47         0.23 

0.49 

8 

0.56 

0.31 

0.56 

,     15 

0.46         0.227 

0.49 

Weight  of  Air  per  Second. — If  i\"is  the  number  of  horse-power  required,  and 
t;  the  efficiency  of  the  air  engine,  "we  have 


vLd  =  550^Y=  7/53.2686^^3  fl  -  — ^ 


Hence 


550 


N 


53.S 


vTs 


10.325  ■ 


N 


vTs     1 


Pi 


(CX.) 


in  English  measures. 
Or  we  may  place 


yLd  =.  7;  2  c,  {T,  -  r.)  =  ooON, 


whence 


550^  N 


N 


''     ^^^3(1-1)"     .^3(1--) 


(CXI.) 


in  English  measures. 
22 


338  THERMODYNAMICS. 

This  last  expression  is  perfectly  similar  to  Equation  C,  page  339,  for  com- 
plete expansion. 

Y 
Volume  of  the  Cylinder. — The  volume  of  the  cylinder  is  ^  ,  «-  denoting  the 

number  of  revolutions  per  second.     From  the  formula 


we  obtain 

V  =         —  — 

2n         2   nrj  (p^  —  lu)  nrj  (ps  —  p^) 

Or  we  may  place 

2n       Pi  ' 


.,_F3       550  i\r.        _^^^  N 


v'  =  ^  =  S-b''^ 


3.  INCOMPLETE  EXPANSION". 

When  the  air  is  cut  oflE  at  such  a  point  that  the  pressure  at  the  end  of  the 
stroke  is  still  greater  than  that  of  the  atmosphere,  we  have  to  deal  with  three 
sets  of  conditions,  viz., 

1.  The  initial  conditions,  p^,  V3,  and  T3.  '      . 

2.  The  conditions  at  the  end  of  stroke,  p^',  V4',  T4'. 

3.  The  conditions  at  exhaust,  p^,  T^,  V^. 

Disposable  WorJc. — The  disposable  work  is  the  work  at  fuU  pressure,  Pi  V3, 
plus  the  work  of  expansion  -7  CviT^—  T^),  less  the  work  of  the  back  pressure 
p^_  Vi'.    Hence 

L,=p3Y3+^c„{T3-T,')-p,V,'. 


This  expression  is  not  convenient  for  use,  hence  we  seek  a  theoretical  quantity, 
Tx,  which,  inserted  in  the  expression  for  disposable  work  for  complete  expansion, 
will  give  the  same  value  for  La  as  would  be  obtained  by  the  expression  above. 
This  is  precisely  similar  to  what  we  have  done  for  full  pressure.  We  obtain  Tx 
from  the  expression 

i>3  F3  +  I  c„  {T3  -  T,')  ^p,  n'  =  1  e,  (n  -  Tx). 


G  T  G  T ' 

Substituting  for  V3  and  F*'  their  values,  -j  {Cp  —  c„)  — ^  and  -j  {Cp  —  c)    ■* 


^  Pi  -^  P\ 


COMPRESSED  AIR  ENGINE— INCOMPLETE  EXPANSION.    339 

and  divided  through  by  -j  ,  we  obtain 

{Cp  -  c,.)  T,  +  c.  {T,  -T,')-  (c,  -  c.)  T,'^^  =  Cp  {T,  -  T.), 


or  by  reduction 


whence 


c,T^'  +  {Cp  -  c„)  T^  ^  Cp  T^, 


^^c^^e^^c.^       1       ^^    .     .     .     (CXII.) 
T^        Cp  Cp    j)i'      k  k    Pi'  ^ 

This  expression  is  the  same  as  CVIII.,  found  on  page  335,  for  full  pressure. 

Tx 

Therefore  the  values  given  in  the  table  on  page  335,  for  ^^  hold  good  here  also 

for  ^. 

We  have  then 

La  =  ^cAT.-T.)^^CpT.,(l-^) 


and 


La  =  ^CpT,(l-^^^ (CXIII.) 


A 


T.'         Tx 

The  values  of  ^  and  7=^  are  easily  found  from  the  tables  on  pages  334  and 

335,  when  the  ratio  ^^  and  ^^  are  given. 

Weight  of  Air  per  Second  and  Volume  of  Cylinder, — From  the  formula 

^        G     ,^       ^^     550iY 

Ld  =  -rCp{Ti  —  Ti) ~ we  obtam 

A.  rj 

in  English  measures. 

This  expression  agrees  exactly  with  that  given  on  page  329  for  complete  ex- 
pansion. 

For  the  volume  of  the  cylinder  we  have 

2n  Pi' 


3J:0  THEBM0DTNAM1C8. 

If  ia  this  expression  we  substitute  for  Q  the  values  just  deduced,  we  get 


Cp        T/n  p^'{Ti  -  T-y 


This  reduces  to 


y'  =  s7o*-i^ 


h      ITU       ,  T.,    /,       Tr\  ' 


F  =  79.75^  ^ (CXV.)^ 


'^''  p,'^.(l-4^ 


Graphic  Representation  of  the  Theory  of  Compressed  Air  Engines. — The  gen- 
eral expression  for  the  aA-ailable  work  of  an  air  engine,  which  includes  all  three 
cases  of  complete  expansion,  full  pressure,  and  incomplete  expansion,  is  then  the 
Eq.  CXIII.,  found  on  page  339,  viz., 

a     ^    ( ,       T,    T^ 

or  in  English  measures 

id  =  183.356^^3  (l  -  ^  ^)  •     •     .     .     (CXVI.) 
We  have  shown  that 

^  =.0.71+ 0.29  fi, 
and 

Substituting  these  values  we  obtain 


=  188.3o(?T3[l-(o.71  +  0.39|^)(^^)    '    J, 


whence 


^A .  =  1  -  (oM  +  0.29 -^^  {^]    ^    .   .     (CXVII.) 

35GT,  \  PUJ\P,I  ^ 


183.356^5^3 


*  [This  expression,  Mr.  Willis  observes,  corresponds  to  that  given  bj'  Mons.  Pernolet,  p.  64 
of  "  Uair  comp?-ime,  but  the  latter  is  defective  in  that  it  contains  no  factor  representing  the  effi- 
ciency of  the  air  engine.] 


COMPRESSED  AIR  ENGINE— GRAPHIC  REPRESENTATION.    341 

we  have 

2/  =  1  -  (0.71  +  0.29.r)  4r (CXVIII.) 


which  is  the  equation  of  a  surface  whose  co-ordinates  are  x,  y,  and  z. 

If  in  this  equation  we  suppose  z  known,  and  therefore  ^^  =  a  constant  =  C, 
we  have 

2/  =  1  -(0.71  +  0.29x-)C, 

which  is  the  equation  of  a  straight  line. 

Hence  if  we  give  to  z  various  vahies  from  1  up  to  any  desirable  limit,  we  shall 
get  a  corresponding  number  of  equations  of  straight  lines,  in  which 

Ld 

y~  im.mGT 

depends  upon  the  disposable  work  and 

Pi 


Since  z=  —.,    and    a;  =  ^  ,  we  have 
Pi  Pi 

z        Pi 

-  =-— ,    or 
X     2h 

P^^'-Pi (CXIX.) 

Now  p^  is  the  pressure  of  the  exterior  air  =  1  atmosphere.  Hence  the  initial 
pressure 

Pj  =  -  atmospheres. 

Let  us  assume  that  we  have  given  the  number  of  disposable  foot-lbs.  of 
work  required  per  second,  the  weight  G  of  air  to  be  used  per  second,  and  the 
initial  temperature  T,,.     From  these  data  we  can  calculate  y. 

If  now  we  have  a  diagram  upon  which  are  drawn  the  lines  whose  equations 
f  re  of  the  form 

2/  =  1- (0.71  +  0.29x)C, 

Vie  can  read  off  from  it  the  values  of  x  and  z  that  will  give  the  desired  value  of 
y.     In  other  words,  we  can  find  at  once  the  initial  pressure  -  in  atmospheres,  and 


342  THERMODYNAMICS. 

the  degree  of  expansion  z  =  =^~  that  will  give  the  required  power  with  the  given 

weight  of  air  per  second. 

Such  a  diagram  has  been  constructed  (see  diagram  at  end  of  this  chapter),  in 

which  the  values  of  z  or  =^,  range  from  1  up  to  10. 


Construction  of  the  Diagram. — Let  the  plane  of  the  pajser  upon  which  the 
diagram  is  drawn  be  a  horizontal  plane,  whose  vertical  ordinate,  referred  to  the 
same  origin  as  the  surface,  is  2  =r  1.  The  equation  of  the  surface,  when  z  is  con- 
stant, that  is,  when  -— ,  is  given,  is  the  equation  of  a  straight  line, 

y  =  l-{0.71  +  0.29a;)  C. 

If  now,  in  this  equation,  we  fix  the  value  of  x,  and  calculate  the  correspond- 
ing value  of  y,  we  can  construct  a  point  of  this  line.  By  finding  two  such  points 
we  can  draw  the  line. 

The  most  convenient  values  for  a;  are  :c  =  0  and  x  =  l.     These  give  us 

fora;  =  0,       y^l-~  =  l '^'^'l_,  ^1-0.71^, 


and 


Now  for  any  given  values  of  z  or  ^  we  can  find  -yfr  from  the  table  on  page 
pi  ^  i 

171,  under  the  head  of  ^,  and  thus  can  easily  calculate  y. 

The  second  result  is  given  by  the  intersection  of  the  horizontal  line  0.28  with 
the  line  z  =2.     The  value  of  x  is  0.61.     Hence 


^3  =  —  =  -7j-pT-  =  3.27  atmospheres. 

That  is,  if  the  cut-off  is  such  that  the  pressure  is  reduced  by  expansion  to  \,  the 
engine  will  give  50  H.  P.  with  one  pound  of  air  at  an  initial  pressure  of  3.27 
atmospheres. 

The  third  intersection  gives  us  3  =  3  and  x  =  0.95,  hence 


,       0.71 

=1-    »■". 

1   — 

or  a:  =  1,       y- 

2"' 

'  1, 

p,i  =  —  =  8.15  atmospheres. 


There  are,  of  course,  many  other  degrees  of  expansion,  with  their  correspond- 
ing initial  pressures,  intermediate  between  those  just  found,  that  will  give  the 


Diagram  for  C(^/7fjoressed  air  I'riff/ie 


Y 

'      . 

' 

', 

? 

7 

■ 

^      /o 

> 

{' 

1 
1 

1 

1 

2 

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^ 

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^^ 

1' 

"- 

^ 

^ 

^ 

^ 

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^ 

-\ 

Z 

\ 

^ 

V 

^^^^ 

^^^ 

^^ 

^ 

z. 

^ 

./ 

i/ 

^\ 

^-^ 

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^^" 

^^ 

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^ 

^-^ 

^ 

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^--k 

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z 

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^ 

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^"-\ 

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(Oz 

^ 

^^ 

^^ 

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"^^^ 

^^\ 

■'z' 

/ 

/ 

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~^ 

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^-\ 

> 

<^^ 

^x^ 

l:::^.;::- 

^\;,_^ 

^^^ 

h' 

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'^A 

-^^^ 

<^^ 

^"\ 

~-^\ 

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7z 

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"^--^ 

"^ 

d/ 

P^ 

-^ 

^>^ 

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^ 

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Oz 

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k. 

i?  / 

X 

X. 

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^^^ 

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^ 

^- 

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w 

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ill' 

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/ 

■ — -— 

^y 

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y^ 

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r 

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x^ 

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^z'  \ 

1 

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x.p  P" 

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m. 

5<,r, 

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— 

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t 

z^    '. 

^ 

Pj--^ 

cttnws. 

X 

^ 

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■e    z=/ 

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i 

Par  CO) 

nf)Le(^eK^ 

'crjisUi/L  . 

/>/ 

f,-^  « 

hnos. 

1 
i 

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■o/x^ 

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1  1  r 

P'a/J(es  c 

/> 

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4;—: 

=^ 

■JT  .0  .7  J  J  K.     '       3.      ^     '^.      6.      7.      8. 


COMPRESSED  AIR  ENGINE— BIAO RAM. 


343 


required  result,  but  which  do  not  appear  on  the  diagram  because  the  diagonal 
lines  corresponding  to  the  ratio  —  are  not  drawn. 

In  order  that  the  initial  pressure  may  2- 

be  found  for  any  degree  of  expansion, 
the  curves  zBC  and  x-^B'C"  are  intro- 
duced. 

These  are  obtained  as  follows  : 

If  we  intersect  the  surface  whose  equa- 
tion is 

y  =  l-{0.n  +  0.29x)^ 

by  any  plane  parallel  to  the  plane  YOZ, 
X  is  for  that  intersection  a  constant,  and 
the  equation  of  the  curve  of  intersec- 
tion is 

2/ =  1  -  (0.71  +  0.39a)  —  . 


The  values  of  y  can  then  be  laid  off  upon  the  vertical  lines  Xj  Y  and  X,  Y'  in 
the  diagram,  and  the  straight  lines  EE'  drawn  through  each  two  points  which 
have  a  common  value  for  Z. 


For  each  of  these  lines  zz'  there  is  a  certain  given  value  for 


.P^ 


hence 


each  line  corresponds  to  a  given  degree  of  expansion,  which  is  denoted  by  the 
figures  1,  2,  3,  etc.,  written  at  the  intersections  of  the  lines  with  x^y'. 

To  illustrate  the  use  of  this  portion  of  the  diagram,  let  us  assume  a  case. 

Given  an  engme  of  50  H.  P.  which  is  to  use  1  poimd  of  air  per  second.  Re- 
quired the  amount  of  expansion  and  initial  pressure  which  will  give  this  re- 
sult. 

We  have  La  =  550  x  50  =  27500  ft.-Ibs.,  and  we  may  assume  T-^  -  527.4° 
Fahr.     We  have  then 


L.i 


rsoo 


183.35GT,        183.35 


527.4 


Find  0.28  on  the  line  X„  Y,  and  following  it  horizontally  across  the  diagram, 
we  can  determine  three  methods  of  arriving  at  the  desired  result. 

The  horizontal  line  first  intersects  the  diagonal  line  for  ^  =r  1,  or  the  line  of 
full  pressure,  at  a  point  whose  abscissa  is  a;  =  0.032.     Hence 


0.032 


31.25  atmospheres. 


That  is,  if  the  engine  works  at  full  pressure  and  uses  1  pound  of  air  per  se- 
cond, the  air  must  have  a  pressure  ^3  =  31.25  atmospheres  to  give  50  H.  P. 


344  THEBM0DTNAMIC8. 

If  now  we  make  a;  =  a  =  0,  and  x  —  a'  =  1,  we  shall  get  the  equations  of  two 
intersections  of  the  surface,  viz., 


for  x  =  0, 


0.71 


for  a;  =  1,        y  —  1 


If  we  revolve  the  intersecting  plane  about  its  line  of  intersection  with  the 
drawing  paper,  the  curve  of  intersection  with  the  surface  takes  the  positions  zBG 
anda;,5'C". 

In  this  position  we  can  construct  them,  remembering  that  since  the  plane  of 
the  drawing  paper  is  assumed  to  have  a  vertical  ordinate  2  =  1,  we  must  lay  off 
2  —  1  instead  of  z  in  the  direction  x^x-^  from  a;,,  and  x-^. 

If  now  we  have  any  value  for  z,  whose  diagonal  line  is  not  given  in  the  dia- 
gram, we  seek  the  points  on  the  curves  EBC  and  x-^B'C  which  correspond  to 
this  desired  z,  and  project  these  points  upon  the  lines  x^^y  and  x-^y'.  Connecting 
these  two  projections,  we  have  the  desired  diagonal  line,  and  may  then  proceed  to 
find  the  initial  pressure  that  will  give  any  desired  power  with  this  amount  of 
expansion. 

It  may  be  required  to  find  the  initial  pressure  which  will  give  some  desired 
power  at  complete  expansion. 

In  this  case  ^4'  —Pi,  and  therefore 


P^  -1: 


Pa 
hence 


Having  found,  therefore,  from  given  data,  the  value  of  y,  for  instance,  in  the 
example  assumed,  y  —  0.28,  we  follow  the  corresponding  horizontal  line  across  to 
its  intersection  with  the  curve  x^B'C,  for  in  the  equation  of  this  curve  x  =  l. 
The  value  of  z  for  the  point  of  intersection  is  read  off  on  the  bottom  line  of  the 
diagram,  and  we  havepj  —  z  atmospheres,  or,  in  the  assumed  example,  js.,  =  3.12 
atmospheres. 

The  case  may  arise  in  which  we  have  given  the  desired  power,  the  weight  of 
air  per  second,  and  the  initial  pressure,  required  the  amount  of  expansion. 


"We  have_p;5  =  — -,  hence  z  =p.iX. 


If  therefore  we  can  draw  upon  the  diagram  curves  of  constant  initial  pressure, 
from  which  for  a  given  y  we  can  readily  find  x,  z  is  easy  to  calculate. 

Such  curves  are  drawn  upon  the  diagram.     To  construct  them,  we  take  the 


COMPRESSED  AIR  ENGINE— DIAGRAM.  345 

equation  x  =  — ,     For  instance,  assume  p^  =  10  atmospheres,  and  give  to  z 

various  values  from  1  up  to  10. 

For  each  value  of  z  we  have  a  value  of  x,  which  we  can  fix  on  the  diagonal 
corresponding  to  that  z.  Through  the  points  thus  obtained  we  draw  the  curve. 
If  now  we  desire  to  know  the  degree  of  expansion  that  will  give  50  H.  P.  from  1 
pound  of  air  per  second  at  an  initial  pressure  of  10  atmospheres,  we  have  as 
before  y  =  0.28.  Following  this  horizontal  line  to  its  intersection  with  the  curve 
of  constant  initial  pressure's  =  10  atmospheres,  we  get  x  —  0.113,  hence 

Z  =  ^^P:,X=:  1.13. 


There  are  one  or  two  points  connected  with  the  diagram  to  which  it  is  well  to 
call  attention. 

A  small  variation  in  the  value  of  x  makes  considerable  difference  in  the  degree 
of  expansion.  That  portion  of  the  diagram  upon  which  x  is  read  off,  is  therefore 
drawn  on  a  scale  twice  that  assumed  for  z.  For  the  same  reason  the  scale  upon 
which  y  is  measured  is  very  much  greater  than  either  of  the  others.  The  scale 
for  measurement  on  the  curve  EBG  is  the  same  as  that  for  XxB'G',  and  is  found 
at  the  top  of  the  diagram. 

Variations  of  Worlc  loitli  Different  Degrees  of  Expansion.— B^eterring  to  the 
diagram,  we  see  that  the  curves  of  constant  initial  pressure  have  a  horizontal 
tangent  at  their  intersections  with  the  line  of  complete  expansion  x^y',  where 
x  =  l,  and  that  their  curvature  is  slight  for  some  distance  from  this  point.  Since 
y,  the  representative  of  the  disposable  work,  is  measured  vertically,  we  see  that 
the  loss  of  work  is  not  very  great  when  the  expansion  is  not  complete  within 
certain  limits.  Assume,  for  instance,  that  we  have  an  engine  working  with  com- 
plete expansion  and  initial  pressure  =  10  atmospheres.  We  find  y  =  0.487. 
Assume,  now,  that  with  the  same  initial  pressure  we  cut  off  at  such  a  point  that 

z=c^-5.     Then  a;  =  —  =  O.C.     Seeking  on  the  diagram  for  the  curve  of 

P*  Pi 

constant  initial  pressure  ,^3  =  10,  the  value  of  y  for  the  abscissa  x=0.5,  we  find 
y  =  0.463. 

The  loss  of  work  is  then  0.487  -  0.413  =  0.05  per  cent.  The  point  of  cut-off 
is  readily  found  from  the  equation 


Referring  to  table  on  page  171,  we  find  for  ^  =  5,  — !-  =  0.319. 

That  is,  cutting  off  at  about  ^  the  stroke,  we  lose  but  0.05  per  cent,  of  the 
work. 

Influence  of  the  Vapor  of  Water  Contained  in  the  Air.— The  preceding  dis- 
cussion has  been  based  on  the  assumption  that  the  air  acted  upon,  or  acting,  was 
dry,  and  followed  the  laws  of  perfect  gases. 


346  THEEMO  DYNAMICS. 

In  practice  water  is  always  injected  into  the  compressing  cylinder,  to  keep 
down  the  final  temperature,  and  the  working  medium  is  therefore  a  mixture  of 
water  with  air  saturated  with  vapor  of  water.  M.  Pemolet  has  discussed  at 
length  the  influence  of  water  and  its  vapor  upon  the  work  of  compression,  the 
disposable  work  of  the  compressed  air,  the  final  temperature  in  the  compressor 
and  the  air  engine,  etc.     He  concludes  from  his  discussion, 

1.  That  the  influence  of  the  vapor  of  water  upon  the  work  of  compression,  as 
well  as  upon  the  disposable  work  of  the  compressed  air  is  relatively  slight,  and 
can  be  neglected  in  all  approximate  calculations  for  which  the  formulae  deduced 
for  dry  air  are  sufficiently  correct. 

2.  Vapor  of  water  in  the  compressor  and  in  the  air  engine  reduces  materially 
the  final  temperature  of  compression,  and  raises  the  final  temperature  of  expan- 
sion. 

The  assumption  that  compression  and  expansion  take  place  according  to  the 
adiabatic  law  is  also  not  strictly  true,  because  the  walls  of  the  cylmder  are  always 
either  receiving  heat  from  or  imparting  heat  to  the  air. 

An  analysis  of  this  question  by  M.  Mallard  shows  that  the  quantity  of  heat 
thus  transmitted  is  small,  and  may  be  neglected. 

Thus,  for  all  practical  purposes,  the  formulae  deduced  for  dry  air  acting  adia- 
batically  are  sufficiently  accurate. 


EXAMPLES    FOE    PEACTICE. 

1.  If  the  specific  heat  of  mercury  under  constant  pressure  is  0.033,  how  many- 
pounds  at  the  temperature  of  240"  will  be  necessary  to  raise  12  pounds  of  water 
from  50'  to  58'  ?  Ans.  16  lbs. 

2.  Let  IV  be  the  weight  of  one  body,  t"  its  temperature,  and  c  its  specific  heat. 
Let  w'  be  the  weight  of  a  second  body,  i'°  its  temperature,  and  c'  its  specific 

heat. 

"What  is  tlie  temperature  x  of  the  mixture  ?  Ans.  x  = ,  ,  - . 

^  wc  +  wc' 

3.  Reduce  —  40"  Fahrenheit  to  Centigrade.  Ans.  —  40'. 
Reduce  -  273'  C.  to  Fahrenheit.                                                Ans.  -  459.4. 

4.  How  do  you  reduce  generally  Fahrenheit  to  Centigrade  degrees  ? 

Ans.  Jf  C.  =  F.  -  32. 

5.  What  outer  work  is  performed  when  1  kilogram  of  air  is  heated  under  the 
pressure  of  the  atmosphere  from  0'  to  1    C  ?  Ans.  29.272  meter-kil. 

6.  What  outer  work  is  performed  when  1  pound  of  air  is  heated  under  the 
pressure  of  the  atmosphere  from  0'  to  1 '  F.  ?  Ans.  53,299  foot-lbs. 

7.  If  the  specific  heat  for  air  under  constant  volume  is  0.16844,  and  under 
constant  pressure  0.23751,  what  is  the  mechanical  equivalent  of  heat  in  French 
measures  ?  Ans.  424  meter-kil. 

8.  If  10  cubic  feet  of  air  are  heated  from  0'  to  10'  C,  what  is  the  new  vol- 
ume ?  J.ns.  10.367  cubic  feet. 

9.  What  is  the  density  ?  Ans.  0.9646. 

10.  If  two  kilograms  of  water  are  heated  under  atmospheric  pressure  from  0' 
to  100°  C,  what  expenditure  of  work  is  equivalent  to  the  heat  imparted  ? 

Ans.  84800  meter-kil. 

11.  What  is  the  weight  of  3  cubic  meters  of  air  at  atmospheric  pressure  and 
20'  C.  temperature  ?  Ann.  3.615  kilograms. 

12.  What  is  the  volume  of  2  kilograms  of  air  at  the  temperature  of  100'  C, 
and  pressure  of  2  atmospheres  ?  Ans.  1.0565  cubic  meters. 

13.  What  is  the  pressure  of  4  kilograms  of  air,  when  the  volume  is  2  cubic 
meters  and  temperature  200"  C.  ?  Ans.  27691.3  kil. 

14.  What  is  the  temperature  of  8  kilograms  of  air,  when  the  pressure  is  6 
atmospheres  and  volume  2  cubic  meters  ?  Ans.  256.55'. 

15.  What  work  is  performed  by  10  kilograms  of  air  at  2  cubic  meters  volume, 
and  5  atmospheres  pressure,  when  it  expands  to  0  cubic  meters,  overcoming  an 

347 


348  THEBMODTNAMICS. 

outer  pressure  equal  at  any  moment  to  the  tension,  the  temperature  being  kept 
constant  ?  Ans.  114210  meter-kil. 

What  is  the  constant  temperature  during  expansion  ?  Ans.  80    C. 

What  is  the  final  pressure  ?  Ans.  1.6G  atmos. 

How  much  heat  must  have  been  imparted  during  expansion  in  order  to  keep 
the  temperature  constant  ?  Ans.  269.4  heat  units. 

16.  If  2  kilograms  of  air,  having  the  volume  of  3  cubic  meters,  expands  as 
above,  performing  work,  from  the  pressure  of  4  atmospheres  down  to  one  atmos- 
phere, and  the  temperature,  during  expansion,  remains  constant,  what  is  the 
work  done  ?  Ans.  171805  meter-kil. 

What  is  the  constant  temperature  during  expansion?  Ans.  1845'  C. 

What  is  the  final  volume  ?  Ans.  12  cubic  meters. 

How  much  heat  must  have  been  imparted  during  expansion  ? 

Ans.  405.2  heat  imits. 

17.  If  a  blowing  engine  changes  per  second  10  cubic  feet  of  air,  at  a  pressure 
of  23  inches,  into  a  blast  at  a  pressure  of  30  inches,  what  is  the  work  per  second? 

Ans.  1366.4  foot-lbs.,  or  189  meter-kil.  per  sec,  or  about  2.5  horse-power. 
What  is  the  volume  after  compression  of  the  10  cubic  feet? 

Ans.  9.33  cubic  feet. 
If  the  temperature  is  60"  F.,  what  is  the  weight  of  10  cubic  feet? 

^MS.  0.72734  lbs. 
How  much  heat  must  have  been  abstracted  during  compression? 

A71S.  1.77  heat  units  per  second. 

18.  If  one  kilogram  of  air  is  heated  under  the  pressure  of  the  atmosphere 
from  0   to  1'  C,  how  much  work  does  it  perform  during  expansion? 

Ans.  29.272  meter-kil. 

19.  If  one  pound  of  air  is  heated  under  the  pressure  of  the  atmosphere  from 
32''  to  33'  Fahrenheit,  how  much  work  does  it  perform  during  expansion? 

A71S.  53.268  foot-lbs. 

20.  What  is  the  weight  of  one  cubic  foot  of  air  at  atmospheric  pressure  and 
32 '  F.  temperature  ?  Ans.  0.08073  lbs. 

21.  What  is  the  weight  of  one  cubic  meter  of  air  under  the  same  pressure  and 
0'  C.  temperature  ?  Ans.  1.29318  kil. 

22.  If  under  the  piston  of  a  steam  engine,  whose  diameter  is  16  inches,  there 
is  a  quantity  of  steam  15  inches  high  and  at  a  tension  of  3  atmospheres,  and  if 
this  steam  in  expanding  moves  the  piston  25  inches,  what  is  the  work  done,  if 
we  assume  Mariotte's  law  to  be  true  for  the  expansion  of  steam?  and  what  is  the 
mean  force  upon  the  piston  when  we  neglect  friction  and  the  opposing  pressure? 

Ans.  10866  foot-lbs.    5217  pounds. 

23.  If  a  given  weight  of  air,  say  2  kilograms,  at  a  temperature  of  30'  C.  ex- 
pands adiabatically,  performing  work,  till  its  volume  is  doubled,  what  is  the  final 
temperature  ?  Ans.  —  44.9°. 

What  is  the  original  volume?  Ans.  1.72  cubic  meters. 

What  is  its  final  pressure  if  the  initial  pressure  is  1  atmosphere? 

Ans.  0.38  atmosphere. 
What  work  does  it  perform?  Ans.  10714.65  meter-kilograms. 

24.  What  is  the  C.  equivalent  of  15'  F.  ?  Ans.  -  9.444. 

25.  Zinc  boils  at  1304'  F.,  mercury  at  608'  F.     Change  these  readinsjs  to  C. 

Ans.  650"  C.  and  320'  C. 


EXAMPLES  FOB  PRACTICE.  349 

26.  Change  the  following  readings  :  Polished  steel  is  of  a  deep  blue  color  at 
580"  F.  ;  polished  steel  is  of  a  pale  straw  color  at  460'  F.  ;  sea- water  freezes  at 
2S^  F.  A71S.  304.5^  C. ;  237.75°  C. ;  -  2.2   C. 

27.  At  0"  C.  a  cast-iron  beam  is  12  feet  long.  What  is  its  length  at  1000'  C, 
supposing  the  law  of  similar  increments  to  hold  for  that  temperature? 

Ans.  12.13524  feet, 

28.  At  25°  C.  a  bar  of  wrought  iron  was  16  feet  long.  What  is  its  length  at 
96°  C.  ?  Ans.  16.01408  feet. 

29.  By  how  much  would  the  length  of  a  submarine  copper  cable  at  0°  shorten, 
if  the  temperature  became  -  20°  C.  ?  Ana.  0.0000434  of  the  length  at  0°. 

30.  A  wheel  of  wrought  iron  has  an  inside  diameter  of  5  feet  when  at  the 
temperature  of  970'^  C.     What  is  its  diameter  at  0    ?  Ans.  4.9  feet. 

31.  A  cylindric  plug  of  copper  just  fits  into  a  hole  4  inches  in  diameter  in  a 
piece  of  cast  iron.  After  heating  the  mass  to  the  temperature  of  1240%  by  how 
much  is  the  diameter  of  the  hole  too  small  for  the  plug  ?     Aiis.  0.0293  inches. 

32.  Two  rods,  one  of  copper  the  other  of  iron,  measure  9.8  decimeters  each  in 
length  at  0°.     What  is  their  difEerence  in  length  at  57   ? 

Ans.  0.0027  decimeters. 

33.  The  wooden  pattern  of  a  cast-iron  beam  must  be  longer  than  the  casting 
at  0°.  For  a  beam  12  meters  long  at  0°,  what  is  the  length  of  pattern  ?  Cast 
iron  melts  at  1530'  C.  Ans.  12.207  meters. 

34.  What  amount  of  work  is  involved  in  lifting  70  lbs.  6  feet  high  ? 

Ans.  420  foot-pounds. 

85.  What  work  is  inA'olved  in  lifting  9000  cubic  feet  of  water  46  feet  ?  (A 
cubic  foot  of  water  weighs  64.4  lbs.)  Ans.  25833600  foot-pounds. 

How  many  units  of  heat  (English)  does  this  correspond  to  ?        Ans.  33463. 

36.  What  work  is  involved  in  a  piston  moving  6  feet  under  an  effective  pressure 
of  17  lbs.  jDer  sq.  inch,  its  area  being  1670  sq.  inches  ?  Ans.  170340. 

37.  The  piston  of  a  steam  engine  is  21  inches  in  diameter,  stroke  6  feet,  mean 
pressure  16  lbs.  per  sq.  inch  ;  the  engine  makes  40  revolutions  per  minute. 
What  is  the  horse-power  (English)  ?  Ans.  80.6144. 

38.  What  time  will  be  taken  by  a  steam  engine  of  64  H.  P.  to  lift  5860  tons 
of  water  20  feet  ?  Ans.  114  minutes. 

39.  How  many  heat  units  (English)  per  hour  are  involved  in  the  idea  of  62 
H.  P.  ?  Ans.  88912.82. 

40.  A  cubic  mile  of  water  is  to  be  lifted  from  a  depth  of  2  feet  in  800  hours. 
How  many  H.  P.  of  a  steam  engine  is  necessary  ?  A  cubic  foot  of  water  weighs 
62.4  lbs.  Ans.  11597.6  H.  P. 

41.  The  resistance  of  friction,  etc.,  to  a  train  is  a  force  equivalent  to  the 
weight  of  600  lbs.  How  many  H.  P.  of  the  locomotive  will  draw  the  train  at  the 
rate  of  35  miles  an  hour  ?  Ans.  56  H.  P. 

42.  How  many  cubic  feet  of  water  will  an  engine  of  10  H.  P.  raise  from  a 
depth  of  150  feet 'in  24  hours  ?  Ans.  50770  cubic  feet. 

43.  What  work  is  performed  on  a  train  weighing  500  tons  in  3  miles  of  a  level 
road,  the  resistance  to  traction  being  -aj,  th  of  the  load  ?  If  this  work  were  done 
in  6  minutes,  what  would  be  the  H.  P.  of  the  engine  ? 

Ans.  65736000  foot-pouncls.    332  H.  P. 


350  THERMOBYNAMIGS. 

44.  Suppose  the  resistance  to  the  progress  of  a  vessel  weighing  1260  tons  to  be 
18  lbs.  a  ton  when  the  speed  is  6  knots,  and  that  the  resistance  varies  as  the 
square  of  the  speed  ;  what  work  will  be  done  by  a  vessel  in  5  nautical  miles,  and 
what  will  be  the  H.  P.  of  the  engine  when  the  speed  is  12  knots  V  There  are  6080 
feet  in  a  nautical  mile,  and  a  knot  is  a  velocity  of  one  nautical  mile  per  hour. 

Ans.  About  2757894000  ;  3342.88  H.  P. 

45.  Wliat  amount  of  work  will  be  spent  in  the  friction  of  a  weight  of  6  tons 
dragged  along  a  level  table  for  a  length  of  7  feet,  when  the  coefficient  of  friction 
is  0.235  ?  What  will  be  the  H.  P.  of  an  engine  which  would  do  this  work  in  one 
second  ?  Ans.  22108.8.   40.19  H.  P. 

46.  In  a  cylinder  we  have  2  kilograms  of  air  at  a  tension  of  Ig-  atmospheres 
and  a  temperature  of  30"  C.     What  is  the  volume  of  this  air  ? 

Ans.  1.14436  cubic  meters. 
If  this  air  expands  adiabatically  till  the  tension  is  1  atmosphere,  what  will  be 
the  final  temperature  ?  Ans.  —3.4'. 

What  will  be  the  final  volume  ?  Ans.  1.5258  cub.  meters. 

What  work  does  it  perform  ?  A7is.  4771.6  meter-kil. 

How  many  units  of  heat  disappear  ?  Ans.  11.2537  heat  units. 

47.  If  3  cubic  meters  of  air  at  150^  and  a  pressure  of  4  atmospheres  expands 
adiabatically  to  double  its  former  volume,  what  is  the  final  temperature  ? 

Ans.  77.6°. 
What  is  the  weight  of  this  air  ?  Ans.  10  kil. 

What  is  the  work  performed  ?  Ans.  51716  meter-kU. 

48.  If  10  kilograms  of  air  occupy  a  space  of  2  cubic  meters,  under  a  pressure 
of  6  atmospheres,  what  must  be  the  temperature  ?  Ans.  150.64°. 

If  it  expands  adiabatically  till  the  final  temperature  is  48%  what  is  the  final 
volume  ?  Ans.  3.94  cub.  meters. 

What  is  the  final  pressure  ?  Ans.  2.3  atmos. 

What  is  the  work  done  ?  A7is.  73316.8  meter-kil. 

49.  Two  kilograms  of  air  at  10°  is  heated  under  constant  atmospheric  press- 
ure till  the  temperature  is  80°.     What  is  the  initial  volume  ? 

Ans.  1.603  cub.  meters. 
What  is  the  final  volume  ?  Ans.  2  cub.  meters. 

What  is  the  work  done  ?  Ans.  4098  meter-kil. 

What  is  the  heat  imparted  ?  Ans.  33.25  heat  units. 

Of  this  heat  how  much  disappears  as  outer  work?         A7is.  9.66  heat  units. 

50.  Four  kilograms  of  air  at  20°  C.  and  under  atmospheric  pressure  is  heated 
and  expands  isopiestically.  After  expanding  till  its  volume  is  doubled,  what  is 
its  temperature  'i*    What  was  its  initial  volume  ?  and  final  volume  ? 

Ans.  313°.  3.32  and  6.64  cub.  meters. 

What  is  the  work  done  ?  Ans.  34306.8  meter-kil. 

How  many  units  of  heat  are  imparted  ?  A^is.  278.4  heat  units. 

How  many  disappear  as  outer  work  ?  Ans.  81  heat  units. 

51.  If  the  air  had  not  been  allowed  to  expand,  and  still  the  same  amount  of 
heat  had  been  imparted,  what  would  have  been  the  temperature  ?  what  the  press- 
ure ?  Ans.  431.34° ;  2.4  atmospheres. 

52.  If  one  kilogram  of  air  has  the  temperature  30°  and  the  pressure  of  1.5 
atmosphei-es,  and  is  cooled,  the  volume  remaining  constant,  till  the  pressure  is 
one  atmosphere,  what  is  the  final  temperature  ?  Ans.  —  71°. 

What  is  the  amount  of  heat  abstracted  ?  Ans.  17.015  heat  units. 


EXAMPLES  FOR  PRACTICE.  351 

53.  If  one  kilogram  of  air  has  the  temperature  30',  and  is  heated  under  con- 
stant pressure  till  the  final  volume  is  ^  of  the  initial,  what  is  the  final  tempera- 
ture ?  Ans.  131^ 

What  is  the  amount  of  heat  imparted  ?  Ans.  23.9885  heat  units. 

What  is  the  outer  work  ?  Ans.  2956.5  meter-kil. 

54.  Suppose  one  kilogram  of  air  of  the  temperature  30"  expands,  according  to 
the  law^,V|'  =p.2V.2~.  until  its  final  volume  is  ;^-  of  its  initial.  What  is  the 
final  temperature  ?  Ans.  —  A^.lb". 

What  is  the  outer  work  ?  Ans.  2217.35  meter-kil. 

What  is  the  specific  heat  in  this  case  ?  Ans.  0.09940. 

Must  heat  be  added  or  subtracted  during  this  expansion  ? 

Ans.  Subtracted. 
How  much  ?  Ans.  7.530  heat  units. 

55.  One  kilogi-am  of  air  of  one  atmosphere  pressure,  30"  temperature,  expands 
adiabatically,  without  overcoming  any  outer  pressure  till  its  volume  is  doubled. 
What  is  the  final  pressure  ?  Ans.  0.5  atmosphere. 

56.  Suppose  10  units  of  heat  are  added  during  expansion.  What  is  the  final 
temperature  ?  Ans.  79.35°. 

What  is  the  final  pressure  ?  Ans.  0.581  afcmos. 


KEDUCTION     TABLES. 


TABLES  POR  THE   CONYERSION  OF  ENGLISH  AND  METRIC   UNITS. 


1  Meter  =  3.2807  feet. 

1  Foot  =  0  3048  meter. 

1  Liter  (vol.  of  1  kllog.  water)  =  0.2202  gal. 

1  Gallon  (vol.  of  10  lbs.  water)  =  4.541  liters. 

1  Kilogram  =  2.204ii  lbs.  av. 

1  Kilog.  per  sq.  meter  -  0  2040  lbs.  per  sq.  ft. 

1  Kilog.  per  sq.  mm.  =  1422.28  lbs  per  sq.  inch. 

1  Lb.  per  sq.  in.  =  703.0958  kilog.  per.  sq.  meter. 

1  Gram  =  15.4323  gr. 

1  Grain  =  0.0648  gram. 

1  Meter-kilogram  =  7.2331  foot-lbs. 


1  Foot-pound  =0.1383  meter-kilog. 

1  Atmosphere  =  14.7  lbs.  per  sq.  in.  =10334  kilog. 
per  sq.  meter  =29.922  inches,  or  760  mm.  of 
mercury  =-33.9  ft.,  or  lOi  meters  of  water. 

1  Pound  av.  =0.4536  kilog. 

Deg.Cent.  =  t  (F.°-32). 

Dcg.  Fahr.  =  ^  (C.°  -t-  32). 

1  Calorie  (kilog.  water  raised  1°  C.)  =  424  meter- 
kilog.  =  3.9683  Eng.  heat  units. 

1  Eng.  heat  unit  (lb.  water  raised  1°  F.)  =  722  ft.- 
Ibs.  =0.252  calorie. 


TABLE  I. 

FOR  CONVERTING   METERS  INTO  INCHES. 


Meters. 

0 

1 

2 

' 

4 

' 

6 

r 

8 

9 

0 

0 

39.4 

7S.7 

118.1 

157.5 

196.8 

236.2 

275.6 

31,5.0 

354.3 

10 

393.7 

433.1 

472.4 

511.8 

551.2 

590.5 

629.9 

708.7 

748.0 

20 

787.4 

826.8 

860.1 

905.5 

944.9 

984.2 

1023.6 

1063.0 

1102.4 

1141.7 

30 

1181.1 

1220.5 

1259.8 

1299.2 

1338.6 

1377.9 

1417.3 

1456.7 

1496.1 

1535.4 

40 

1574.8 

1614.2 

1653.5 

1732.3 

1771.6 

1811.0 

18.50.4 

1889.8 

1929.1 

50 

1968.5 

21107.9 

2047.2 

2086.6 

2126.0 

2165.3 

3204.7 

2244.1 

2283.5 

2322.8 

60 

2401.6 

2440.9 

2480.3 

2519  7 

2.O59.0 

2598.4 

2637.8 

2677.2 

2716.5 

70 

2755.9 

2795.3 

2834.6 

2874.0 

2913.4 

2952.7 

2992.1 

3031.5 

3070.9 

3110.2 

80 

3149.6 

3189.0 

3228.3 

3267.7 

3307.1 

3346.4 

3425.2 

3464.6 

3303.9 

90 

3513.3 

3582  7 

3622.0 

3661.4 

3700.8 

3740.1 

3779.5 

3818.9 

3858.3 

389T.6 

TABLE  IL 

FOR,  CONVERTING   INCHES   INTO   CENTIMETERS. 


Inches. 

0 

1 

2 

3 

4 

5 

6 

r 

8 

9 

0 

0 

0.254 

0  508 

0.762 

4.016 

1.2699 

1,5239 

1.7779 

20319 

2  28.59 

1 

2.5400 

2.7940 

3.0480 

3.3020 

3.5,560 

3.8099 

4.06.39 

4.3179 

4..5719 

4.8259 

2 

5.0799 

5.3339 

5.,58T9 

5.8419 

6,0959 

6.3498 

6.6038 

6.8578 

7.1118 

7.36,58 

a 

7.6199 

7.8739 

8.1279 

8  3S19 

8,63,59 

8.P898 

9.1438 

9.3978 

9.6518 

9,90.58 

4 

10.1599 

10.4139 

10.6679 

10.9219 

11.1759 

11.4298 

11.9378 

12.1918 

12  44,58 

5 

12.699.'^ 

12.9538 

13.2078 

13.4618 

1.3.71,5S 

13  9(;97 

14.22-37 

14.4777 

14.7317 

14,98.57 

6 

15.2398 

15.4938 

15.7478 

16  0018 

16.25,58 

16.5097 

16.76.37 

17.0177 

17.2717 

17.5257 

7 

17.7798 

18.0338 

18.2878 

18.5418 

18.79,58 

19.0497 

19.3037 

19.5,577 

198117 

20.0657 

8 

20.3197 

20.5737 

20.8277 

21.1817 

21.3,367 

21.5896 

21.8436 

22.1976 

22..3516 

22.60.56 

9 

22.8.597 

28.1137 

23.  .3677 

23.6217 

23.8757 

24.1296 

24.3836 

24.6376 

24.8916 

25.1456 

10 

25.3997 

25.6537 

25.9077 

26.1617 

26.41.57 

26.6696 

26  9236 

27.1776 

27.4316 

27.6856 

11 

27.9396 

28.1936 

28.4476 

28.7016 

28.9556 

29,2095 

29.4035 

29.7175 

29.9715 

30.2255 

352 


BEBTJCTION  TABLES. 


353 


TABLE  III. 

FOR   CONVERTING   FRENCH   MEASURES   INTO   ENGLISH. 


Meter, 

sq.  m., 

Feet. 

Inches. 

Sq.  ft. 

Sq.  in. 

Cub.  ft. 

Cub.  iu. 

cubic  ra. 

1 

3.2800 

39.3706 

10.7642 

15.50.05 

35.3161 

61026.2 

2 

6.5618 

78.7412 

21.5284 

3100.09 

70.6322 

1220.52.4 

3 

9.8427 

118.1118 

32.2936 

4650.13 

105.9483 

18.3078.6 

4 

13.1235 

157.4824 

43.0568 

6200.18 

1-11.2644 

244104.9 

5 

16.4044 

196.8530 

53.8201 

77.50.23 

176.5805 

305131.1 
366157.3 

6 

19.68.53 

236.9237 

64.585-2 

9300.27 

211.8966 

22.9662 

275.2943 

75.3494 

10850.31 

247.2126 

427183.5 

8 

26.2471 

314.9649 

86.1136 

12400.36 

2S2.5287 

488209.7 

9 

29.5280 

354.3355 

96.8778 

1.3950.40 

317.8448 

5492.35.9 

TABLE  IV. 

FOR  CONVERTING  ENGLISH   MEASURES   INTO   FRENCH. 


Foot, 
sq.  ft., 
cub.  ft. 

Meter. 

Sq.  m. 

Cub.  m. 

Inch, 
^q.  in  , 
cub  in. 

1 

Centi- 
meters. 

Sq  cent. 

Cub.  centi- 
meter.«. 

1 

0..304796 

0.092901 

0.028316 

2.5400 

6.4514 

16.386 

2 

0.61 19592 

0.18.5801 

0.056631 

2 

5.0799 

12.9029 

32.773 

3 

0.914388 

0.27870SJ 

0.08^947 

3 

7.6199 

19.3543 

49.1.59 

4 

1.219184 

0.371602 

0.113:^63 

4 

10.1599 

25.80,57 

65.546 

5 

1.523979 

0.464503 

0.141579 

5 

12,6998 

32.2.571 

81.9.32 

() 

1.828775 

0.5.57403 

0.169894 

6 

15.2398 

38.7086 

98.31 H 

7 

2.133571 

0.650.304 

0.198210 

17.7799 

45.1600 

114.705 

8 

2.4.^8367 

0.74.3204 

0.226526 

8 

20.3197 

51.6114 

131.091 

9 

2.743163 

0.836105 

0.254841 

9 

S2.8597 

58.0628 

147.478 

10 

3.047959 

0.929005 

0.283157 

10 

25.3997 

64.5143 

163.864 

11 

3  352755 

1.021906 

0.311473 

11 

27.9396 

70.9657 

180.250 

TABLE  V. 

FOR  CONVERTING   KILOGRAMS   INTO  POUNDS  AVOIRDUPOIS,  OR  CALORIES  INTO  ENGLISH 
(CENTIGRADE)   HEAT  UNITS. 


Kilo- 
grams. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0 

2.205 

4.409 

6.014 

8.818 

11.023 

13.228 

15.432 

17.637 

19.841 

10 

22.046 

24.251 

26.4.55 

28.660 

30.864 

83,0t)9 

35.274 

37.478 

39.683 

41.887 

20 

44.092 

46.297 

48.501 

.50.706 

52.910 

5.5.115 

59.524 

61.729 

63.933 

30 

66.138 

68.-343 

70.547 

72.752 

74.956 

77.161 

79.366 

81.570 

83.775 

85.979 

40 

b8.184 

90.389 

92.593 

94.798 

97.002 

99.207 

101.412 

10.3.616 

105.821 

108.025 

50 

110.230 

112.435 

114.639 

116.844 

119.048 

121.253 

123.458 

125.662 

127.867 

130.071 

60 

132.276 

134.481 

136.6S5 

138.890 

141.094 

143.299 

145.504 

147.708 

149.913 

1.52.117 

70 

154..322 

156.527 

1.58.731 

160.936 

163.140 

165..345 

167.5.50 

169.754 

171.959 

174.163 

80 

176  368 

178.573 

180.777 

182.982 

185.186 

187.391 

189..596 

191.800 

194.005 

196.209 

90 

198.414 

200.619 

202.823 

205.028 

207.232 

209,437 

211.642 

213.846 

216.051 

218.255 

23 


354 


THEBMOD  TNAMICS. 


TABLE  VI. 

FCS   CONVERTING  AVOIRDUPOIS   POUNDS   INTO  KILOGRAMS,  OR  ENGLISH   (CENTIGRADE) 
HEAT   UNITS    INTO    CALORIES. 


Pounds. 

0      1 

2 

3 

4 

5 

G      7 

8 

9 

0 

0 

0.4536 

0.9072 

1.3608 

1.8144 

2.2680 

2.7216   3.175i 

3  6288 

4.0824 

10 

4.536 

4.9896 

5.4432 

5.8968. 

6.3504 

6.8040 

7.2.576   7.7112 

8.1648 

8.6184 

20 

9.072 

9..5256 

9.9T92 

10  43iy 

10.8864 

11.3400 

11.7936  12.2472 

12.7008 

13.1544 

30 

13.608 

14.0616 

14.5152 

14.9688 

15.4224 

15.8760 

16.3296  16.7832 

17.2.368 

17.6904 

40 

18.144 

18.5976 

19.0512 

19.5048 

19.9584 

20.4120 

20.8656  1  21.3192 

21.7728 

22.2264 

50 

22.680 

23.1336 

23.5Br2 

24.0408 

24.4944 

24.9480 

25.4016  25.8553 

26.3088 

26.7624 

60 

27.216 

27.6696 

28.1232 

28.5768 

29.0304 

29.4840 

29.9376  30.3912 

30.8448 

31.2984 

70 

31.752 

32.20.56 

32.6592 

33.1128 

33.5664 

34.0200 

34.4736  34.9272 

35.3808 

35.8344 

80 

36.288 

36.7416 

37.1952 

37.6488 

38.1024 

38  5560 

39.0098  39  4632 

39.9168 

40.3704 

90 

40.824 

41.2776 

41.7312 

42.1848 

42.6384 

43.0920 

43.5456  43.9992 

44.4528 

44.9064 

TABLE  VII. 

FOR   CONVERTING   METER-KILOGRAMS   INTO   FOOT-POUNDS. 


Meter 
kilog. 

0 

1 

2 

3 

4 

5 

6 

r 

8 

9 

0 

0 

0.72331 

1.44662 

2.16993  2.89324 

3.61655 

4.33986 

5.06317 

5.78648 

6.50979 

1 

7.2331 

7.9564 

8.6797 

9.4030  !  10.1263 

10.8496 

11.5729 

12.2962 

13.0195 

13.7428 

2 

14.4662 

1.5.1895 

15.9128 

16.6361  17.3,594 

18.0827 

18.8060 

19.5293 

20.2526 

20.9759 

3 

21  6983 

22.4226 

23.14,59 

23  8692  24.5925 

25.31.58 

26.0391 

26.7624 

27.4857 

28.2090 

4 

28.9324 

29.6.5.57 

30.3790 

31.10-23  31.82.511 

32.  .5489 

33.2722 

33.9955 

34.7188 

35.4421 

5 

36.1655 

36  8838 

37.6121 

38.33.54  39.0587 

39.7S20 

40.5053 

41.2286 

41.9519 

42.6752 

6 

43.3986 

44.1219 

44.8452 

45.5685  ;  46.2918 

47.0151 

47.7384 

48.4617 

49.1850 

49.9083 

7 

50.6317 

51.3550 

52.0783 

.52.8016  53.5249 

54.2482 

54.9715 

55.6948 

56.4181 

57.1414 

8  ' 

57.8648 

58.5881 

593114 

60.0,347  i  60.7580 

61.4813 

6i.2046 

62.9279 

63.6512 

64.3745 

9 

65.0979 

65.8212 

66.5445 

67.2378  67.9911 

68.7144 

70.1610 

70.8813 

71.6076 

TABLE  Vlll. 

FOR   CONVERTING   FOOT-POUNDS  INTO   METER-KILOGRAMS. 


Foot 
lbs. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0 

0.01.383 

0.02766 

0.04149 

.05,532 

.06915 

.08291 

.09681 

.11064 

.j2447 

1 

0.1.383 

0.1521 

0.1659 

0.1797 

0.19.35 

0.2073 

0.2211 

0.2349 

0.2487 

0.2625 

0.2766 

0.2904 

0.3042 

0.3180 

0..3318 

0.3456 

0..3594 

0.3732 

0.3870 

0.4008 

3 

0.4149 

0.4287 

0.4425 

0.4,563 

0.4901 

0.4839 

0.4977 

0.5115 

0.5253 

0.5391 

4 

0..5532 

0.5670 

0.5S08 

0..5946 

0.6084 

0.6222 

0.6360 

0.6498 

0.6636 

0.6774 

5 

0.6915 

0.70.53 

0.7191 

0.7329 

0.7467 

0.7605 

0.7743 

0.7881 

0.8019 

0.8157 

C 

0.8298 

0.8436 

0.8574 

0.8712 

O.8850 

0.8988 

0.9126 

0.9264 

0.9402 

0.9540 

7 

0.9681 

0.9819 

0.99.57 

1.0095 

1.0233 

1.0371 

1.0.509 

1.0647 

1.0785 

1.0923 

8 

1.1064 

1.1202 

1.1,340 

1.1478 

1.1616 

1.17.54 

1.1892 

1.2030 

1.2168 

1.2306 

9 

1.2447 

1.2585 

1.27-'3 

1.2861 

1.2999 

1.3137 

1.3275 

1..3413 

1.3551 

1.3689 

REDUCTION  TABLES. 


355 


TABLE  IX. 

FOR   CONVERTING   KILOGRAMS   PER   SQ.    CENTIMETER   INTO    POUNDS   PER   SQ.    INCH. 


0 
142.228 
284.45fi 
426.684 
508.912 
711.140 


14.2228 
156.4.11 
298.679 
440.907 
583.135 
725.363 
867.591 
1009.819 
1152.047 
1294.275 


28.4456 
170.674 
312.902 
455.130 
597.358 
739.586 
881.814 
1024.042 
llOti.970 
1308.498 


241.788 
384017 
526.245 
668.473 
810.701 
9.52.929 
1095.167 
12.37.385 
1379.613 


113  7824 
256  Oil 
398.240 
540.468 
682.696 
824.924 
967.152 
1109390 
1251.608 
1393  836 


19S.005a 
270.234 
412.463 
554  691 
696.919 

981.375 
1133.613 
1265.831 
1208.059 


TABLE  X. 

FOR   CONVERTING   POUNDS   PER   SQ.    INCH  INTO  KILOGRAMS   PER   SQ.    CENTIMETER. 


Pounds 

per 

0 

1 

2 

3 

4 

5 

6 

7 

8 

sq.  inch. 

0 

0 

.070309 

.140618 

.210937 

.381236 

.351545 

.421854 

.492163 

.562472 

10 

.70309 

.77340 

.84371 

.91402 

.98433 

1.05464 

1.12495 

1.19.526 

1.26557 

20 

1.40618 

1.47649 

1.54680 

1.61711 

1.68742 

1.75773 

1.82804 

1.89835 

1  9li666 

30 

2.1092? 

2.17958 

2.24989 

3  33030 

2.39051 

2.46082 

2.53113 

2.60144 

3  67175 

40 

2.81236 

2.88267 

2  95298 

3.02329 

3.09360 

3.16391 

3.23422 

3.304.-3 

3.37484 

50 

3.52545 

3.58.576 

3.65607 

3.72638 

3.79669 

3.86700 

3.93731 

400763 

4.07793 

60 

4.21854 

4  28885 

4.35916 

4.42947 

4.49978 

4.57009 

4.64040 

4.71071 

4.78102 

70 

4.92163 

4.99194 

5.06225 

5.132.56 

5.20287 

5.27318 

5.34349 

5  41380 

5.48411 

80 

5.62472 

5  69503 

5.76534 

5.83565 

5.90596 

5.97627 

6.04658 

6.11689 

6.18720 

90 

6.32781 

6.39812 

6.46843 

6.53874 

6.00905 

667936 

6.74967 

6.81998 

6  89029 

TABLE  XL 

FOR   CONVERTING   ATMOSPHERES   INTO   POUNDS  AND  KILOGRAMS. 


Atmospheres. 

in  can 

1 

76 

3 

152 

3  

238 

304 

5     . 

380 

6 

456 

533 

8  

9 

608 
684 

Height  of  mercury 
in  inches. 


29.992 
59.844 
89.766 
119.688 
149.610 
179.532 
209.454 
239.376 
269.298 


Pressure  per 
sq.  cent.  sq.  inch 

n  kOosrrs.       in  pounds. 


1.0333 
2.0666 
3.0999 
4.1332 
5.1665 


14.696 

5i9.393 
44.088 
58  784 
73.480 


103.872 
117.568 
132.264 


356 


THEBMOD  TNAMIC8. 


TABLE  XII. 

FOR    CONVERTING    CALORIES    INTO    ENGLISH    (FAHRENHEIT)    HEAT    UNITS. 


Calor- 
ies. 

0 

1 

2 

3 

4 

5 

6 

1 
7   1   8 

9 

0 

0 

3.969 

7.938 

11.907 

15,876 

19.745 

23.814 

27.783  i  31.752 

35.721 

10 

39.69 

.43.659 

47.628 

51.597 

55.566 

59.535 

63.504 

67.473  t  71.442 

75.411 

20 

79..38 

83.349 

87.318 

91.287 

95.256 

99.225 

103.194 

107.163  111.1.32 

115.101 

30 

119.07 

123.0.39 

127.008 

130.977 

134.946 

138.915 

142.884 

146.853  :  150.822 

154.791 

40 

158.76 

162.729 

166.698 

170.667 

174.636 

178.605 

182.574 

186.543  1  190.512 

194.481 

50 

197.45 

201.419 

205.388 

209.357 

213.326 

217.295 

221.264 

225.233  ;  289.202 

233.171 

60 

238.U 

242.109 

246.078 

250.047 

254.016 

257.985 

261.954 

265.923  1  269.892 

273.861 

70 

277.83 

281.799 

285.768 

289.737 

293.706 

297.675 

301.644 

305.613  i  309.582 

313.561 

80 

317.52 

331.489 

325.4.58 

329.427 

333.396 

337.365 

341.334 

345.303  349.272 

.353.241 

90 

357.21 

361.179  j  365.148 

369.117 

373.086  j  377.055 

S81.024 

384.993  1  388.962 

1 

392.931 

TABLE  XIII. 

FOR    CONVERTING    ENGLISH    (FAHRENHEIT)    HEAT    UNITS    INTO    CALORIES. 


Heat 

Uuits. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0 

0.252 

0.504 

0.7.56 

1.008 

1.260 

1.512 

1.764 

2.016 

2.268 

10 

2.52 

2.772 

3.024 

3.276 

3.528 

3.780 

4.032 

4.284 

4.536 

4.788 

20 

5.04 

5.292 

5.544 

5.796 

6.048 

6.300 

6.552 

6.804 

7.056 

7.308 

30 

7.56 

7.812 

8.064 

8.316 

8.568 

8.820 

9.072 

9.324 

9..576 

9.828 

40 

10.08 

10.332 

10.584 

10.836 

11.088 

11.340 

11.592 

11.844 

12.096 

12.348 

50 

12.60 

12.852 

13.104 

13.356 

13.608 

13.860 

14.112 

14.364 

14.616 

14.868 

60 

15.12 

15.372 

15.624 

15.876 

16.128 

16.380 

16.632 

16.884 

17.1.36 

70 

17.64 

17.892 

18.144 

18.396 

18.648 

18.900 

19.152 

19.404 

19.6.56 

19,908 

80 

20.16 

20.412 

20.664 

20.916 

21.168 

21.420 

21.672 

21.924 

22.170 

22.428 

90 

22.68 

22.932 

23.184 

23.436 

23.688 

23.940 

24.192 

24.444 

24.948 

THERMODYNAMICS 


PART    SECOND. 


STEAM   AKD    THE    STEAM   ENGINE 


CHAPTER   XIV. 

THE  ACTION    OF  HEAT    IN    EVAPOEATION. — GENEEAL    PROPERTIES   OF 
STEAM. — PRESSURE    OF   SATURATED    STEAM. 

The  different  Effects  of  Heat  in  Evaporation. — Of  all  the  investi- 
gations thus  far  made  upon  the  deportment  of  bodies,  none  are 
of  greater  importance  than  those,  which  relate  to  the  properties 
of  steam.  An  exact  knowledge  of  these  properties  lies  at  the 
foundation  of  the  construction  of  the  steam  engine. 

Although  much  in  the  way  of  investigation  was  done  from  the 
time  of  Watt,  who  gave  us  the  steam  engine  almost  in  its  pres- 
ent state  of  perfection,  down  to  the  time  of  Kegnault,  one  of 
the  most  skillful  experimenters  of  the  present  day,  still,  much 
had  to  remain  unknown,  because  during  that  time  the  princi- 
ples of  the  mechanical  theory  of  heat  were  either  unknown  or 
but  little  regarded. 

It  will  not  be  difficult  to  call  attention  to  that  which  must 
during  that  time  have  remained  hidden,  the  key  to  the  explana- 
tion of  which  is,  however,  given  by  a  number  of  phenomena  of 
great  scientific  and  technical  interest.  Let  us  return  to  the 
experiment  which  we  have  already  described  in  Chapter  YIIL 
of  Part  First. 

Again,  let  ABCD,  Fig.  58,  be  a  hollow  cylinder  of  1  sq.  meter 
cross-section.  IT23on  the  bottom  CD  is  1  cubic  decimeter =0.001 
cubic  meter  =  1  liter  of  water,  whose  weight  is  therefore  1  kilo- 
gram at  0°.  In  contact  with  the  surface  of  the  water  is  the 
piston  KK,  which  we  assume  perfectly  tight  and  without  weight. 
Above  the  piston  is  a  vacuum. 

We  assume  the  piston  loaded  with  10334  kilograms.  This  is 
the  same  as  the  pressure  of  the  atmosphere  per  square  meter 
for  0°  and  760  millimeters  (30  inches)  barometric  height. 

Now  let  us  impart  heat  to  the  water  until  the  temperature 

359 


360 


TIIEBMOD  TNAMIC8. 


B 


rises  to  100^.  Up  to  this  point  no  steam  is  formed.  This  only 
happens  when  the  addition  of  heat  is  carried  still  farther.  The 
piston  will  then  be  gradually  raised,  as  steam  is  formed,  and 
the  weight  10334:  kilograms  will  be  lifted  through  a  certain  dis- 
tance. When  all  the  water  is  converted  into  steam, 
the  piston  will  stand  at  a  height  of  about  1.650 
meters  above  the  bottom. 

In  this  apparently  simple  process  we  can  recog- 
nize, from  the  standpoint  of  the  mechanical  theory 
of  heat,  several  different  effects. 

First,  the  water  is  heated,  therefore  the  vibration 
ivork  of  the  molecules  is  increased.  When  the 
water  attains  the  temperature  of  100°,  further  rise 
of  temperature  does  not  occur.  The  velocity  of  the 
molecules  is  now  so  great,  however,  that,  according 
to  our  theoretical  views,  any  further  addition  of 
heat  not  only  separates  the  molecules  beyond  the 
influence  of  their  cohesion,  which  constituted  the 
liquid,  but  also  gives  them  a  rectilinear  motion 
such  as  the  molecules  of  a  gas  possess.  The  mole- 
K  Y  ''"|^^""jK  cules  then  impinge  upon  the  piston  and  raise  it, 
IJ  C  until  all  the  water  is  converted  into  steam.     For 

this  transformation  of  the  water  into  steam,  this 
overcoming  of  the  forces  which  bind  the  molecules  of  the  liquid 
mass  together,  or  " disgregation  loorh"  there  is,  as  we  know, 
a  certain  amount  of  heat  necessary. 

Further,  the  weight  of  10334  kilograms  is  raised  about  1.65 
meters.  This  requires  a  mechanical  work  of  10334  x  1.65  = 
17051  meter-kilograms,  and  this  work  must  evidently  be  also 
furnished  by  the  heat.  Since  a  work  of  424  meter-kilograms  is 
equivalent  to  one  heat  unit,  the  work  of  17051  meter-kilograms 

requires     .^^r^  =  40.2  heat  units.    At  the  end  of  the  experiment, 

the  heat  actually  existing  in  the  steam  as  heat,  or  energy  of 
vibration,  must  then  be  less  than  the  total  amount  imparted, 
by  just  this  amount,  which  is  required  for  the  performance  of 
outer  work.  It  is  this  circumstance  to  which  we  wish  here  to 
call  special  attention.  From  it  flow  a  number  of  otlier  facts  as 
to  the  deportment  of  steam,  which  withoiit  it  either  cannot 
be  recognized  or  else  are  wrongly  explained.      Thus,  for  ex- 


ACTION  OF  HEAT  IN  EVAPORATION.  361 

ample,  tlie  steam  used  per  stroke  by  the  steam  engine  possesses 

at  the  end  of  the  stroke  no  longer  the  heat  which  it  had  in  the 

boiler,  and  the  heat  which  thus  disappears  is  the  equivalent 

of  the  work  done.     This  has  been  verified,  as  we  have  seen,  by 

Hirn's  experiment.     If  the  water  had  expanded  greatly,  while 

heated  from  0°  to  100° — which,  as  we  know,  is  not  the  case — 

then   the  outer  work  of  10334  x  1.65   meter-kilograms  would 

have  been  partly  performed  by  this  expansion. 

If  we  compare  the  outer  work  of  17051  meter-kilograms,  which 

we  may  call  the  outer  work  of  steam,  with  that  necessary  for 

overcoming  the   molecular  forces,  we  shall  find  considerable 

difference.      Thus  in  the  present  case,  196.3  heat  units  are 

necessary  for  this   purpose    alone.      This   heat   represents   a 

mechanical  Avork  of  496.3  x  424  =  210431  meter-kilograms,  and 

this  is  required  to  overcome  the  molecular  forces  of  only  one 

kilogram  of  water.     Indeed,  in  comparison,  the  force  of  gravity 

is  but  slight.     If,  for  example,  a  mass  of  water  of  one  cubic 

meter  =  1000  kilograms   is  required  to  perform  this  work,  it 

210431 
must  sink  through  a  distance  of  -:.^„„    =  210.431  meters. 

The  question  arises,  How  can  the  heat  be  determined  which 
is  necessary  for  the  different  effects  in  this  transformation  of 
water  into  steam  ?  It  is  not  possible  to  determine  with  exact- 
ness the  heat  imparted  during  the  heating  and  vaporization 
of  the  water,  by  direct  determination  of  the  heat  furnished  by 
the  fuel.  The  opposite  method,  of  determining  the  heat  units 
set  free  when  the  steam  condenses,  is  far  more  exact.  If  we 
use  for  the  condensation  a  large  quantity  of  cold  water,  the 
condensation  is  rapid,  and  all  the  heat  is  given  up  to  the  water, 
while  the  amount  lost  by  radiation  and  conduction  is  very 
slight.  If  we  use  2545  kilograms  of  water  at  0°,  which  is  raised 
in  temperature  1°,  while,  the  temperature  of  the  condensed 
steam  is  also  diminished  ^°,  we  shall  have,  evidently,  the  fol- 
lowing equation  for  the  heat  contained  by  the  steam  : 

a;  -  1  =  i  X  2545, 

where  x  is  the  number  of  units  of  heat  possessed  by  the  steam. 
Hence 

X  —  636.5  heat  units. 


362  THERMODYNAMICS. 

It  is  evident  tliat  in  this  experiment,  the  steam  being  con- 
densed by  cold  water,  that  heat  reappears  which  disappeared 
in  the  raising  of  the  weight.  This  heat  is  therefore  contained 
in  the  total  heat  of  the  water. 

If  we  assume  that  for  the  heating  of  the  water  from  zero  to 
100"  100  heat  units  are  necessary  (as  we  shall  see  hereafter 
this  number  is  somewhat  too  small),  then,  for  the  disgregation 
work  and  the  outer  work,  636.5  —  100  =  536.5  heat  units  are 
required.  Accordingly  the  total  heat  in  our  experiment  is 
divided  among  the  various  processes  as  follows : 

1.  Heating  of  the  water  from  0 '  to  100'  (vibra- 

tion work)    100     heat  units. 

2.  Overcoming  tile  molecular  forces  (disgrega- 

tion work) 496.3     "       " 

3.  Raising  10334  kil.  1.65  meters  (outer  work) .   40.2     " 

Total 636.5     "       " 

In  calling  attention  thus  to  the  circumstances  which,  in 
ignorance  of  the  principles  of  the  mechanical  theory  of  heat, 
must  have  escaped  those  physicists  who  have  investigated  the 
phenomena  of  vaporization,  the  question  arises  whether  their 
determinations  are  therefore  worthless,  or  have  their  experi- 
ments been  so  conducted  that  we  can  make  use  of  them  ?  The 
question  is  to  be  answered  decidedly  in  the  affirmative,  es- 
pecially as  regards  the  comprehensive  and  careful  experi- 
ments of  Regnault,  made  with  large  and  accurate  apparatus, 
described  on  page  367,  and  surpassing  in  accuracy  all  other 
experiments  of  the  same  character  by  other  physicists. 
The  experiments  of  Regnault  upon  steam,  as  also  upon  other 
vapors,  depend  not  only  upon  the  exact  determination  of  the 
heat  necessary  to  raise  1  kilogram  of  water,  under  a  given 
pressure,  from  0"""  to  any  other  temperature,  or  to  convert, 
under  the  same  circumstances,  1  kilogram  of  water  at  0°  into 
steam,  but  also  upon  the  careful  determination  of  the  pressure 
of  the  steam  generated  from  water  at  any  temperature.  It  is 
not  the  23lace  to  describe  in  detail  the  apparatus  used  in  these 
remarkable  experiments,  nor  to  notice  here  the  very  ingenious 
methods  of  investigation.     This  belongs  to  experimental  phys- 


GENERAL  PROPERTIES  OF  STEAM. 


363 


ics.  We  sliall  only  cite  some  of  liis  experiments,  in  order  to 
show  that  they  answer  well  the  purposes  of  our  investiga- 
tions. 

Before  doing  this,  we  shall  call  attention  to  the  principal 
properties  of  steam  and  the  different  kinds  of  steam. 


General  Properties  of  Steam. — Saturated  and  Supei^lieated  Steam. 
— Let  /.,  //.,  ///.,  Fig.  59,  be  prismatic  or  cylindrical  vessels 
of  1  square  meter  cross-section,  each  holding  1  kilogram  of 
water,  the  air-tight  piston  KK  being  in  contact  with  the  sur- 
face of  the  water.  Let  the  piston  in  I.  be  loaded  with  10334, 
that  in//,  with  2  x  10334,  and  mill,  with  3  x  10334  kilograms, 
the  space  above  being  a  vacuum. 

In  order  that  the  j)iston  in  /.  shall  begin  to  rise,  or  vapori- 
zation    begin,    experi- 


ments show  that  the 
water  must  be  heated 
up  to  100^  In  //  the 
water  must  be  heated 
up  to  120.6".  In  ///, 
up  to  134^ 

If  all  the  water  is 
converted  into  steam 
in  all  three  cylinders, 
the  piston  in  /.  will  be  .^r  L_Z___Ji/-  tr 
raised,  as  shown  by 
experiment,  to  1.65  m., 
in  //.  to  0.86,  and  in  ///.  to  0.59  meters, 
kilogram  =  0.001  cubic  meters  of  water 


TI 


m 


K    K 


I 


K 


In  /.,  therefore,  the  1 
has  become  1.65  cubic 
meters,  in  //  the  same  water  volume  becomes  0.86,  and  in  ///. 
0.59  cubic  meters  of  steam.  In  /.,  then,  from  1  cubic  meter  of 
water,  we  should  obtain  1650  cubic  meters  of  steam  of  100° ;  in 
//.,  860  cubic  meters  at  120.6° ;  and  in  ///.,  590  cubic  meters  at 
134".  The  number  which  denotes  how  many  cubic  meters  of 
steam  are  generated  from  1  cubic  meter  of  water,  or  generally, 
the  ratio  of  the  volume  of  the  steam  to  that  of  the  water  from 
which  it  is  generated,  is  called  the  "  specific  steam  volume.'''' 
In  /  it  is  1650 ;  in  //,  860 ;  in  ///,  590. 

Since  in  //.  the  same  weight  of  steam  (1  kilogr.)  is  contained 
in  about  half  the  space  that  it  occupies  in  /.,  the  density  (weight 


364  THERMODYNAMICS. 

of  unit  of  volume)  is  about  twice  as  great.  In  III.  we  iiave  in 
about  Ad  of  the  volume  the  same  weight  as  in  L,  hence  the 
density  is  about  3  times  as  great  as  in  /. 

"We  can  recognize,  then,  the  following  principles  : 

1.  The  greater  the  pressm^e  the  higher  the  temperature  at  ivhich 
vaporization  begins. 

2.  Since  the  steam  holds  the  outer  pressure  in  equilihrium,  the 
higher  the  temperature  the  greater  the  pressure  of  the  steam  generated. 

3.  From  a  certain  volume  of  ivater  there  is  generated  for  any 
given  temperature  a  definite  and  invariable  steam  volume. 

(The  ratio  of  this  volume  to  the  water  volume,  or  the  specific 
steam  volume,  must  be  less  the  greater  the  temperature.) 

4   The  density  is  greater  for  high  temperatures  than  for  loiver. 

5.  The  greater  pressure  of  the  steam  at  high  temperatures  de- 
pends less  upon  the  difference  of  temperature  than  upon  the  greater 
density. 

We  have  here  assumed  that  the  steam  is  formed  directly  from 
the  water,  and  that  steam  and  water  are  in  contact  up  to  the 
moment  when  the  last  drop  of  water  is  evaporated.  Such  steam, 
which  for  reasons  we  shall  soon  learn  we  call  "  satui'ated,"  must 
be  distinguished  from  other  kinds  of  steam  which  we  shall  have 
occasion  to  speak  of. 

If,  when  all  the  water  is  just  converted  into  steam,  and  the 
pistons  are  at  their  highest  positions,  we  fix  the  pistons  im- 
movably and  force  steam  into  the  spaces  already  filled  with 
steam,  condensation  will  occur  and  an  amount  of  steam  will  be 
condensed  equal  in  weight  to  the  amount  of  fresh  steam  forced 
in.  The  same  will  be  the  case  if  the  pistons  are  forced  down, 
provided  the  temperature  remains  constant.  We  say,  there- 
fore, that  the  spaces  are  filled  with  "  saturcded  steam,''  that  is, 
steam  just  at  the  point  of  condensation.  As  long  as  steam  is 
in  contact  with  the  water  from  which  it  is  being  generated,  it 
must  at  any  moment  be  at  its  point  of  condensation,  and  there- 
fore "saturated."  Saturated  steam  differs,  then,  essentially 
from  a  permanent  gas,  in  that  it  cannot  be  compressed  like  gas 
under  constant  temperature.  If  thus  compressed,  a  portion  is 
condensed  while  the  pressure  remains  the  same. 

If,  for  example,  Os,  in  Fig.  60,  is  the  steam  volume  of  1.65 
cubic  meters  which  is  generated  in  /.  from  1  kilogram  of  water, 
and  sB  =  p  is  the  pressure  (10,384  kil.  per  sq.  meter),  and  if  by 


GENERAL  PROPERTIES  OF  STEAM. 


365 


forcing  down  tlie  piston  tlie  steam  is  compressed  gradually, 
then,  under  the  condition  that  the  temperature  is  hept  constant,  tlie 
pressure  p  remains  tlie  same.  If  all  the  steam  is  converted  into 
water,  we  have  finally  one  kilogram  of  water  at  100'',  which  oc- 
cupies the  space  Os^  of  0.001 
cubic  meter.  The  straight  ^ 
line  'BA.,^ parallel  to  OX,  reij- 
resents  tlie  change  of  condition 
of  the  steam  when  compressed 
imder  constant  tempercdure. 
If,  inversely,  we  evaporate 
the  1  kilogram  of  water  of 
100^  under  the  constant 
pressure  p,  the  point  A 
passes  from  A  to  B.  It 
is  evident  that  in  compressing  the  steam  we  must  abstract, 
in  order  to  keep  the  temperature  constant,  as  much  heat  as 
must  be  imparted  during  its  generation. 

If  now  we  consider  the  steam  in  I.  still  further  heated,  then 
if  the  pressure  remains  constant  it  expands  while  its  temper- 
ature increases.  Suppose  that  from  the  moment  of  expansion 
it  is  no  longer  in  contact  with  water.  The  steam  is  now  no 
longer  saturated,  no  longer  just  at  the  point  of  condensation, 
and  since,  with  the  same  pressure,  it  has  a  higher  temperature, 
we  call  it  "superheated." 

Superheated  steam,f  tJien,  is  steam  which  for  the  same  pressure 
has  a  greater  temperature  and  a  greater  specific  volume  than  scdu- 
rcded  steam. 

The  more  the  steam  is  superheated,  the  more  it  approaches 
the  condition  of  a  permanent  gas,  but  only  at  a  very  consider- 
able distance  from  the  point  of  saturation  are  its  properties 
essentially  the  same. 

If  the  j)istons  in  I.,  II.,  and  III.  are  held  fast,  and  then  more 
heat  added,  both  the  temperature  and  pressure  increase.  The 
steam  is  superheated.  If,  for  example,  in  I.  the  pressure  is 
about  twice  as  great,  the  temperature  is  considerably  greater 
than  120.6°,  which  is  that  of  saturated  steam  of  the  same  press- 
ure.    We  also  understand,  therefore,  by  superheated  steam, 


["  Isopiestic  line."] 


t  [Also  sometimes  called  "  steam  gas."] 


366 


THERMOB  TNAMICS. 


st3am  loMcJi/or  the  same  specific  volume  lias  a  higher  pressure  and 
a  higher  temperature  than  saturated  steam. 

Suppose  we  liave  below  tlie  piston  KK,  Fig.  61,  liiglily  super- 
heated steam.  "We  now  force  the  piston  gradually  down.  The 
deportment  of  the  steam  is  at  first  that  of  a  per- 
TT-L ^jg-  manent  gas.  If,  therefore,  we  keep  the  temper- 
ature constant,  the  pressure  increases  inversely 
with  the  volume,  and  the  change  of  condition  fol- 
lows Mariotte's  law.  If  heat  is  not  abstracted,  the 
pressure  increases  in  a  greater  ratio  according  to 
the  exponential  law  of  Poisson,  or  "  adiabatically," 
only  we  have  now  no  longer  the  exponent  1.41, 
which  was  for  air,  but  another  value  for  the  expo- 
nent. The  more  we  force  down  the  piston,  under 
constant  temperature,  the  more  we  approach  a 
point  where  Mariotte's  law  no  longer  holds  good. 
This  point  lies  near  the  point  of  saturation  or  of 
condensation.  When  this  point  is  reached,  further 
compression  simply  causes  condensation,  the  press- 
ure remains  constant  and  Mariotte's  law  entirely 
ceases  to  apply. 

That  which  has  been  said  concerning,  steam  applies  to  all 
"vapors,"  that  is,  to  all  gaseous  bodies  generated  from  liquids, 
and  which  by  ordinary  compression  or  cooling  can  be  recon- 
verted into  liquids.  Such  are  the  steam  from  spirits  of  wine, 
ether,  carbonic  acid,  mercury,  etc.  Only  the  numerical  rela- 
tions between  temperature,  pressure,  specific  volume,  etc.,  are 
different. 

We  pass  now  to  a  subject  which  is  also  of  special  interest 
in  the  mechanical  theory  of  heat,  that  is,  to  the  exact  deter- 
mination of  the  relation  between  the  pressure  of  steam  and  its 
temperature. 


Empirical  Formuloefor  the  Pressure  and  Temperature  of  Sat- 
urated Steam. — ^Very  many  experiments  have  been  made  in 
order  to  determine  the  pressure  of  saturated  steam  at  differ- 
ent temperatures.  None  of  them  possess  greater  reliability 
and  exactness,  and  none  are  more  comjDrehensive  than  those 
made  by  Regnault.  The  method  of  his  investigations  depends 
upon  the  principle  which  holds  good  for  every  liquid,  that  the 


SATURATED  STEAM— BEGjSTAULT'S  EXPERIMENTS.      367 

temperature  at  tvhich  ivater  toils  remains  constant  so  long  as  the 
pressure  upon  the  liquid  is  constant,  and  that,  therefore,  the  press- 
ure of  the  steam  is  in  equilibrium  with  this  pressure.  Experi- 
menters before  Eegnault  measured  the  steam  pressure  direct- 
ly. Now  it  is  very  difficult  to  maintain,  even  for  only  a  few 
minutes,  the  temperature  of  the  water,  and  hence  the  expan- 
sive force  of  the  steam,  constant,  in  a  vessel  closed,  and  ex- 
posed to  fire,  because  the  heat  of  the  fire  varies.  Yet  this  is 
necessary  while  noting  the  temperature  and  pressure.  In  order 
to  avoid  the  inaccuracies  arising  from  such  method  of  observa- 
tion, Regnault  compressed  the  air  which  surrounded  his  vessel. 
In  order  that  the  steam,  when  generated,  should  not  act  to  in- 
crease the  outer  pressure,  it  was  condensed  in  a  special  vessel 
just  as  soon  as  it  was  generated.  We  give  a  sketch  of  Beg- 
nault's  apparatus.  Fig.  62.  C  is  a  small  copper  boiler,  into 
which  four  thermometers  enter  ;  two  enter  the  water  and  two 


the  steam  only,  in  order  to  determine  if  the  water  and  steam 
have  always  the  same  temperature.  The  steam  space  of  C  is 
connected  by  a  pipe,  TT,  with  a  large  globe,  A,  of  about  24 
liters  capacity,  which  is  contained  in  a  vessel,  VV,  filled  with 
water. 


368  THERMODYNAMICS. 

From  tliis  globe  leads  a  pipe  with  cock,  R,  to  an  air-pump, 
so  tliat  tlie  air  can  be  compressed  at  will  in  the  globe  and 
boiler.  The  pressure  of  the  air  in  botJi  globe  and  boiler  is 
shown  by  the  manometer,  nn,  mm,  which  communicates  with 
the  globe  by  the  pipe  Ix.  The  pipe  TT  is  surrounded  by  a 
jacket  rr,  through  which  cold  water  continually  circulates. 
The  steam  generated  in  G  must,  therefore,  since  the  air  is  kept 
cool,  be  at  once  condensed,  and  cannot  therefore  contribute  to 
increase  the  expansive  force  of  the  air.  When,  now,  the  air  in 
A  and  in  C  has  been  compressed,  and  the  pressure  exactly 
noted  on  the  manometer,  the  water  in  C  is  heated  until  the 
thermometer  shows  a  constant  temperature.  This  we  may  be 
sure  is  the  temperature  under  which,  for  the  given  pressure, 
the  water  boils.  And  the  expansive  force  of  the  steam  at  this 
temperature  is  exactly  equal  to  the  pressure  as  indicated  by 
the  manometer. 

Eegnault  used  in  his  experiments  a  large  and  a  small  appara- 
tus. The  first  served  especially  for  the  determination  of  press- 
ures for  temperatures  between  170^  and  230",  the  other  between 
0°  to  170°. 

The  experiments  of  Eegnault  have  thus  far  furnished  no 
exact  law  as  to  the  relation  of  pressure  and  temperature  of 
saturated  steam.  We  have  only  empirical  formulae  which  give 
this  relation  with  more  or  less  accuracy.  Of  all  these  for- 
mulae, none  agree  with  the  results  of  experiment  better  than 
that  of  Eegnault  himself,  as  also  the  formula  of  Magnus  and 
Eontgen. 

Eegnault  gives  three  formulae,  the  first  for  temperatures  from 
-  32°  to  0°  C,  the  other  for  from  0°  to  100°  C,  and  the  third  for 
temperatures  above  100°  up  to  230°  C. 

The  first  formula  has  the  form 

(-32°to0°)  jy  =  a-\-la^ 

where     a  =  -  0.08038,    log  l  =  1.6024724,    log  a  =  0.0333980, 

where  t  is  the  temperature.  This,  as  well  as  all  the  other 
formulae  of  Eegnault,  gives  the  pressure  p  in  millimeters  of 
barometric  height,  760  millimeters  to  one  atmosphere. 


SATURATED  STEAM— REGNAULT' 8  EXP EBIMENT8.      369 
The  second  formula  of  Kegnault  lias  tlie  form 

(0°  to  100°)  log  p  =  a  +  ha*  -  eft' 

wliere  a  =  47393707,  log  h  =  2.131990711,  log  a  =  0.006864937, 
log  c  =  0.611740767,  log  /i  =  1.996725536,  and  t  is  again  the 
temperature. 

The  third  formula  is 

(100"  to  230°)  log.  2^  =  a-ha^-  c/5^ 

where  a  =  6.2640318,  log  6  =  0.1397743,  log  «  =  1.994049292, 
log  c  =  0.6924351,  log  ft  =  1.99834862,  ic  =  ^  +  20. 

Before  giving  other  formulae,  let  us  see  how,  from  the  pre- 
ceding, the  pressure  of  steam  may  be  found  from  the  tem- 
perature. 

EXAMPLE. 

"What  is  the  pressure  of  saturated  steam  whose  temperature  is  20°  C.  ? 

We  must  use  liere  the  second  formula. 

First,  log  at  =  0.006864937  x  20  =  0.13729874  and  log  I  +  log  a'  = 
2^131090711  +  0.13729874  =  2:269289  =  the  number  0.0185903.  We  have  for 
log  fi<,  L 996725536  X  20=1.93451072,  and  log  c  +  log  /3<  r=  0.611740767  + 
193451072  =  0.54625149  =  the  number  3.5177.  Then,  a  -  cfd' =  4.73937  -  3.5177 
=  1.22167.  Finally,  log  p  =  0.0185903  +  1.22167  =  1.24026.  Hence i?  =  17.388 
millimeters.  More  exact  calculation  would  have  given  17.371.  Table  I.  at  the 
end  of  this  work  gives  the  pressure,  according  to  Regnault's  calculations,  from 
-  32°  to  230°. 


The  formula  of  Magnus  is 

>  =  4.525  X  10' 


■4475  < 

4.69 +  « 


in  which  t  is  the  temperature  and  p  the  pressure  in  millimeters. 
It  gives,  for  temperatures  below  100°,  excellent  results  as  com- 
pared with  experiment.  Above  100°  the  agreement  is  not  so 
good.  By  a  change  of  coefficients,  however,  great  exactness 
may  be  attained  here  also.  The  formula  is,  moreover,  very 
convenient  for  calculation,  as  an  example  will  show. 
24 


370  THEBM0DTNAMIC8. 


EXAMPLE. 

What  is  the  pressure  of  saturated  steam  at  130^  ? 

1^475  X  130  958.175 

Here  t=  130,  henee  i?  =  4.525  x  W^*^'^  +  ^^^  =4.525  x  10^^"^  =  4.525   x 
102.6548_  Now  the  log  of  10'^^548  is  2.6548,  and  the  corresponding  number  is  451.65. 

Hence  p  =  4.525  x  451.65  =  2042.716   millimeters  =  ^^^:!^'^  =  2.688    atmos- 

7d0 

pheres.     According  to  Regnault's  formula  we  should  have  2.671  atmospheres. 

The  formula  also  lias  the  advantage  tliat  we  may  find  in- 
versely tlie  temperature  from  the  pressure. 
Thus  we  have 

,  1       ^  trot.      7.4475  X  t 

log^.  =  log  4.525 +  23CT^:rT' 
or 

234.69  log^  +  i^  log^  =  234.69  log  4.525  +  t  log  4.525  +  7.4475^, 

hence 

_  234.69  (log  j3- log  4.525)  ^  234.69  log  j9  -  153.867 
~  log  4.525  -  log  p  +  7M75  8.10312  -  log ^       * 


EXAMPLE. 

If  the  pressure  of  steam  is  lOmm.  what  is  its  temperature  ? 
Here  p  =  10,  hence  log  jJ  —  1,  and  we  have 

_  234.69  -  158.867  _  80.823  _ 

7.103  ^  7.103   -"•^''; 

The  formula  of  Eontgen,  for  temperatures  from  —  32°  to  100°, 

log  p  =  log  760  —  (a  +  hx  +  cx^  +  dx'^)  x, 

where  a  =  0.015432,        h  =  0.0000542,        e  =  0.0000000704, 
d  =  0.0000000000066,        a;  =  100  -  t 

Here  t  is  the  temperature,  and  p  is  given  in  millimeters. 
lip  is  given  in  atmospheres,  we  have  more  simply 

log  j)=  —  {a  +  hx  +  co(^  +  dx*)  a?, 


the  coefficients  being  the  same. 
Above  100°  the  formula  is 


log  p  =  log  760  —  (a  +  hx  +  ca?-)  x. 


SATURATED  STEAM— FORMULA.  371 

c  =  0.0000000704 


Where  a  =  0.015432,        h  =  0.00004265 
and  ic  =  100  -  t 


Here  again,  for  the  pressure  in  atmospheres,  we  have 

log  }}=  —  {a  +  hx  +  CX-)  X. 


EXAMPLE. 

What  is  the  tension  of  saturated  steam  when  the  temperature  is  105.08'  ? 

Here  t  =  105.08,  hence  100  —  t  =  —  5.08.  Therefore,  in  the  equation  log  p  = 
log  760  —  (a  +  bx  +  ex-)  x,  the  first  and  third  term  in  the  parenthesis  will  be 
positive,  and  the  second  negative. 

We  have 

a =      0.015432 

ex'  =  0.0000000704  x  25.806 =       0.00000182 

0.01543382 

bx  =  0.00004265  x  -  5.08 --=  -  0.00021666 

a  +  bx  +  ex-^ =   0.01521716 

-  (a  +  bx  +  c.«2).x- =   0.0773024 

log  760 =       2.8808136 

logji =      2.9581160 

p  =  908.06  millimeters. 

Eegnault  found  by  experiment  904.87  millimeters. 
The  following  table  gives  the  pressures  for  other  liquids,  ac- 
cording to  Eesnault. 


Temperature 
C. 

Ether. 

Alcohol. 

Acetic  Acid. 

Chloroform. 

Chloride  of 
Carbon. 

Bisulphide  of 
Carbon. 

0° 

184.38 

12.70 

63.33 

59.72 

32.95 

127.91 

10 

286.83 

24.23 

110.32 

100.47 

55.97 

198.46 

20 

432.78 

44.46 

180.08 

160.47 

90.99 

298.03 

30 

634.80 

78.52 

280.05 

247.51 

142.27 

443.62 

40 

907.04 

133.69 

419.35 

369.26 

214.81 

617.53 

50 

1264.83 

219.90 

608.81 

535.05 

314.38 

857.07 

60 

1725.01 

350.21 

860.96 

755.44 

447.43 

1164.51 

70 

2304.90 

541.15 

1189.90 

1042.11 

621.15 

1552.09 

80 

3022.79 

812.91 

1611.05 

1407.64 

843.29 

2032.53 

90 

3898.26 

1189.30 

2140.82 

1865.22 

1122.26 

2619.08 

100 

4953.30 

1697.55 

2796.20 

2428.54 

1467.09 

3325.19 

110 

6214.63 

2367.64 

3594.33 

8110.99 

1887.44 

4164.06 

120 

7719.30 

3231.73 

4551.95 

3925.74 

2393.67 

5148.79 

130 

4323.00 

5684.90 

4885.10 

2996.88 

6291.60 

140 

5674.59 

7007.64 

6000.16 

3709.04 

7603.96 

372  THERMODYNAMICS. 


QUESTIONS   FOR   EXAMINATION. 

What  pressure  in  kilograms  per  square  meter  is  equal  to  one  atmosphere  ?  What  is  a  liter  ? 
How  many  millimeters  of  the  barometer  correspond  to  one  atmosphere  ?  When  water  is  heated 
in  a  vessel  under  constant  pressure,  describe  the  different  effects  of  the  heat  imparted.  In  each 
kilogram  of  water,  how  many  heat  units  go  to  perform  the  outer  work  ?  How  many  to  perform 
disgregation  work  ?-  How  many  to  perforin  vibration  work  ?  What  then  is  the  total  heat  im- 
parted in  heat  units  ?    Define  what  you  mean  by  heat  unit. 

What  is  the  relation  between  pressure  and  temperature  at  which  vaporization  begins?  Illus- 
trate. What  is  "specific"  steam  volume?  Illustrate.  How  does  this  volume  vary  with  the 
temperature  ?  How  does  the  density  of  steam  vary  with  the  temperature  ?  Upon  what  does 
the  increased  pressure  of  steam  at  high  temperatures  mainly  depend  ? 

What  do  you  understand  by  saturated  steam?  Illustrate.  How  does  it  differ  from  perma- 
nent gas  ?  What  do  you  mean  by  superheated  steam  ?  Why  is  it  called  *'?/;;er-heated  ?  What  is 
its  specific  volume  as  compared  with  saturated?  For  the  same  specific  volume  how  do  the  press- 
ure and  temperature  compare  with  saturated  ?  If  saturated  steam  Ls  compressed  under  constant 
temperature,  what  happens  ?  If  superheated  steam  is  compressed,  what  happens  ?  What  is  a 
"vapor  ?" 

Upon  what  principle  do  the  experiments  of  Regnault  depend  ?  What  was  the  object  of  them  ? 
Why  cannot  the  steam  pressure  be  measured  directly  ?  How  did  Regnault  avoid  these  inaccura- 
cies ?  Describe  his  apparatus  and  its  method  of  working.  Have  his  experiments  given  any 
exact  law  between  pressure  and  temperature  of  saturated  steam  ?  If  saturated  steam  has  a  cer- 
tain temperature,  has  it  a  definite  pressure  ?  Below  what  limit  of  temperature  does  the  formula 
of  Magnus  give  good  results?  What  limits  are  included  by  Eegnanlt's  formulfe?  Are  these 
formulae  within  their  limits  reliable?  How  were  they  deduced?  Between  what  limits  does 
ESntgen's  formula  hold  good?  Do  any  of  these  formute  hold  good  for  superheated  steam? 
Why  not?    What  is  given  by  Table  I.  ? 


CHAPTEE  XV. 

HEAT  OF  THE   LIQUID. — TOTAL   HEAT. — INNER  AND   OUTER   HEAT   OF 
VAPORIZATION. — HEAT   OF  THE   STEAM. 

Specific  Heat  and  Heat  of  the  Liquid. — Eegnaiilt  lias  also  inves- 
tigated, with  the  same  careful  accuracy,  whether  the  amount  of 
heat  required  by  water  for  a  certain  given  rise  of  temperature 
is  the  same  at  high  temperatures  as  at  low.  In  other  words, 
whether  the  specific  heat  of  water  is  constant  for  all  tempera- 
tures— if,  for  example,  the  heating  of  one  kilogram  of  water 
from  0^  to  10^,  20^,  etc.,  requires  ten  or  twenty  times  as  much 
heat  as  from  0^  to  1°.  Kegarded  from  the  standpoint  of  the 
mechanical  theory  of  heat,  this  involves  the  following  ques- 
tions : 

1.  Is  the  work  required  for  a  certain  rise  of  temperature  of 
the  water  greater  or  less  for  high  temperatures  than  for  low  ? 

2.  Is  the  disgregation  work  and  outer  work  greater  or  less  ? 

This  last  would  be  the  case,  if  the  water  at  high  tempera- 
tures expanded  more  or  less  than  at  low.  Numerous  experi- 
zients  have,  however,  shown  that  the  expansion  of  water  is 
almost  the  same,  relatively,  under  the  ordinary  pressure  of  the 
atmosphere,  for  all  temperatures.  Since,  as  we  have  seen  al- 
ready in  Part  I.,  the  disgregation  work  is  extremely  small  in 
comparison  with  the  vibration  work,  we  can  also  neglect  its 
influence,  and  have  only  to  consider  that  heat  which  is  requi- 
site for  the  rise  of  temperature. 

The  apparatus  used  by  Regnault  consisted  in  part  of  that 
which  he  made  use  of  for  the  determination  of  steam  pressure, 
viz.,  of  a  boiler,  in  which  the  pressure  and  temperature  of  the 
steam  and  water  were  carefully  determined.  By  means  of  a 
pipe,  closed  by  a  cock,  the  water-space  of  the  boiler  was  con- 
nected with  a  calorimeter.     The  water  passing  from  the  boiler 

373 


374  THERMODYNAMICS. 

to  tliis  calorimeter  could  be  exactly  determined,  and,  by  ther- 
mometers, the  temperature  of  the  cooling  water  was  deter- 
mined at  every  instant.  By  a  suitable  apparatus  the  hot  water 
was  made  to  mix  quickly  with  the  cold  water  of  the  calorim- 
eter, so  that  as  little  heat  as  possible  was  lost  by  radiation 
and  conduction. 

After  the  air  pressing  upon  the  water  in  the  boiler  had  been 
compressed  to  a  given  point,  and  the  water  brought  to  the  boil- 
ing point,  the  cock  was  opened  for  a  short  time,  so  that  a  cer- 
tain quantity  of  water  passed  into  the  calorimeter,  under  al- 
most constant  pressure. 

If,  now,  the  water  while  heated  up  to  boiling  had  expanded 
much,  it  would  have  overcome  the  air  pressure  through  a  cer- 
tain distance,  performed  a  certain  amount  of  mechanical  work, 
and  a  certain  quantity  of  heat  would  have  disappeared.  On 
entrance  into  the  calorimeter,  the  water  would  then  have  con- 
tracted under  the  same  pressure,  work  would  have  been  re- 
ceived by  it,  and  heat  would  have  reappeared.  "We  should, 
therefore,  have  recovered  the  heat  necessary  for  the  expansion 
of  the  water.  Further,  the  water  rushes  into  the  calorimeter 
with  a  certain  velocity.  For  the  generation  of  this  velocity  a 
certain  amount  of  heat  must  be  expended.  But  neither  can 
this  be  lost,  because  the  water  comes  gradually  to  rest  in  the 
calorimeter,  so  that  its  living  force  is  transformed  into  heat 
again.  Therefore  the  heat  appearing  in  the  calorimeter  is  pre- 
cisely that  which  the  water  received  in  the  boiler. 

From  his  experiments  Regnault  found  by  calculation  that  the 
specific  heat  of  water  between  the  temperatures  ^i  and  t,  for  ex- 
ample, between  50''  and  40",  or  between  90°  and  30^  is  given  by 
the  equation 

C,,  _,  =  1  +  a  (^1  +  0  +  5  (^1^  +  ttx  +  f)    .     .     (1). 

in  which  ty  is  the  higher  and  t  the  lower  temperature.  For 
from  0°  to  ^1°  we  have,  therefore, 

(7,,  =:  1  +  ah  +  K^ (2). 

In  order  to  determine  the  coefficients  a  and  5,  Begnault 
found  first  the  mean  specific  heat  of  water  between  0°  and  lOO"*, 
or  that  heat  which  in  the  mean  is  required  by  1  kilogram  of 


HEAT  OF  THE  LIQUID.  375 

water  between  0°  and  100°  to  raise  its  temperature  1°.     He 
found  for  this  1.005  lieat  units,  so  tliat  we  have  from  (2) 

Cioo  =  1.005  =.  1  +  a  X  100  +  Z^  X  100^    .     .     (3). 

For  tlie  specific  heat  between  0°  and  200°,  he  found  C^  = 
1.016.     Hence  we  have 

C200  =  1-016  =  1  +  «  X  200  +  6  X  200^    .     .     (4). 

From  (3)  and  (4)  a  and  h  can  be  readily  found.     We  have 

a=  0.00002     and     h  =  0.0000003  ; 

hence  equation  (1)  becomes 

(7,, _,  =  1  +  0.00002  {t  +  fi)  +  0.0000003  {f  +  tt^  +  t,%     (5). 

EXAMPLE. 

What  is  the  mef^n  specitic  heat  of  water  between  15°  and  25°,  and  how  much 
heat  is  necessary  to  raise  1  kilogram  of  water  from  15°  to  25°  ? 
Here  ti  =25,  and  t  =  15,  and  t  +  ti  =40.     Hence 

C~25-u--=l  +  0.00002  X  40  +  0.0000003  x  (15-  +  15  x  25  +  25^) 
=  1  +  0.0008  +  0.0000003  x  1225 
=  1.0011675  heat  units. 

Therefore  the  heat  necessary  to  raise  1  kilogram  of  water  from  15°  to  25°  is 
1.0011675  (25  -  15)  =  1.0011675  x  10  =  10.011675  heat  units.  If  the  specific  heat 
of  water  Avere  constant  for  all  temperatures,  the  heat  required  would  have  been 
simply  10  heat  units. 

If  in  (5)  we  put  0  in  place  of  f,  we  have  the  specific  heat  of 
water  between  0°  and  t°.  If  we  denote  this  mean  specific  heat 
by  C„i,  we  have 

Ah  =  1  +  0.00002^  +  0.0000002^^     .     .     .   (I.) 

Thus  the  mean  specific  heat  between  0°  and  150°  is 

C^  =  l  +  0.00002  X  150  +  0.0000003  x  150^  =  1.00975. 

The  amount  of  heat  necessary  to.  raise  1  kilogram  of  water 
from  0°  to  f  is  evidently 

CJ  =  (1  +  0.00002^  +  0.0000003^2)  i5 
=  t  +  0.00002^2  +  0.0000003^. 


376  THEEMODTNAMICS. 

Tills  lieat  is  very  appropriately  called  tlie  "lieat  of  the 
liquid,"  and  denoted  by  q,  so  that 

q=t  +  0.00002^2  +  0.0000003^^  .    .     .     (II.) 

This  very  important  equation  gives,  therefore,  the  quantity 
of  heat  in  heat  units  necessary  to  raise  1  kilogram  of  water 
from  0"  to  f. 

The  values  in  column  5  of  Table  II.  are  calculated  from  this 
formula. 

Eegnault  has  found  the  heat  of  the  liquid,  for  other  liquids 
also,  and  given  empirical  formulae,  as  follows : 

For 

Ether q  =  0.52901^  +  0.0002959^1 

Alcohol q  =  0.54754^  +  0.001122^^  +  0.000002^. 

Acetic  acid q  =  0.50643«5  +  0.000397^1 

Chloroform q  =  0.23235^  +  0.0000507^1 

Chloride  of  carbon.  ...q  =  0.19798/5  +  0.0000906^^^ 
Bisulphide  of  carbon... (/  =  0.23523^  +  0.000082^1 

We  have  thus  far  spoken  of  the  mean  specific  heat  of  water, 
that  is,  of  the  mean  amount  of  heat  required  to  raise  1  kilogram 
of  water  one  degree,  between  the  limits  t°  and  ^1°  or  0°  to  t". 
But  this  is  not  the  heat  necessary  to  raise  the  temperature  of 
water  1°  above  a  given  point,  or,  more  properly,  a  very  little 
above  that  point.  Since  in  this  case  t^  is  but  little  greater  than 
t,  we  find  the  actual  specific  heat  at  t°  by  making  ti  =  t  in  (5). 

If  we  denote  this  "  actual  specific  heat "  by  C  simply,  we 
have 

(7=1  +  0.00002  (20  +  0.0000003  (3^^)^ 
or  « 

C  =  1  +  0.00004^  +  0.0000009^2 ,    .    .    .     (III.)    • 


Example  1. — What  is  the  actual  specific  heat  of  water  at  100"  and  at  130°  ? 
For  100"'  we  have 

C  rr:  1  +  0.00004  X  100  +  0.0000009  x  10000 
:=  1  +  0.004  +  0.009  =  1.013. 

If,  then,  we  denote  the  heat  necessary  to  raise  1  kilogram  of  water  at  0 '  a  very 


TOTAL  HEAT  AND  HEAT  OF  VAPORIZATION.  377 

little  higher,  by  x,  the  heat  necessary  to  raise  1  kilogram  at  100",  the  same  small 
amount  is  1.013.r. 

The  specific  heat  at  130"  is 

c  =  1  +  0.00004  X  130  +  0.0000009  x  16900 
=  1  +  0.00520  +  0.01521 
=  1.02041. 

Example  3. — How  many  heat  units  are  necessary  to  raise  100  kilograms  of 
water  from  0"  to  100"  ? 
From  (II.)  we  have 

?  =  100  +  0.00002  X  10000  +0.0000008  x  1000000 
=  100  +  0.2  +  0.3  =  100.5  heat  units, 

hence  100  kilograms  will  require,  in  order  to  raise  the  temperature  from  0°  to 
100°,  100.5  X  100  =:  10050  heat  units. 


The  heat  necessary  to  raise  one  kilogram  of  water  from  t  to 
ti  degrees,  is  given  by 

q^-q  =  t,-t  +  0.00002  {f,'  -  f)  +  0.0000003  {t,'  -  f). 

Thus,  to  raise  one  kilogram  of  water  from  10 ""  to  100^,  we 
require 

q,-q  =100 -10+  0.00002  (100^  -  10^)  +0.0000003  (100^  -  10^) 
=  90  +  0.198  +  2997  =  90.4977  heat  units. 

Total  Heat  and  Beat  of  Vaporization. — The  "  total  heat,"  that 
is,  the  amount  of  heat  necessary  in  order  to  raise  1  kilogram  of 
water  gradually  from  0^  to  saturated  steam  of  a  given  tempera- 
ture and  pressure,  has  also  been  determined  with  great  care 
by  Kegnault.  A  part  of  the  apparatus  used  in  these  experi- 
ments consisted,  as  before,  of  the  boiler  and  globe  already 
described.  The  steam  generated  was  conducted  into  a  calo- 
rimeter, which  was  also  connected  with  the  globe.  Since,  there- 
fore, the  water  was  evaporated  under  the  same  pressure  at 
which  it  was  condensed,  the  heat  obtained  in  the  calorimeter 
was  equal  to  that  imparted  to  the  steam  in  the  boiler. 

The  observations  showed  that  the  heat  imparted  to  1  kilo- 
gram of  water,  in  order  to  convert  it,  at  various  pressures,  into 


378  THEBM0DTNAMIC8. 

saturated  steam,  increased  slowly  with  tlie  temperature,  and 
was  given  by  the  following  empirical  formulae  : 

^=606.5  +  0.305^.     .....     (IV.) 

where  W  is  the  total  heat  and  t  the  temperature  of  the  water 
or  steam. 

Thus  in  order  to  convert  1  kilogram  of  water  into  steam  at 
10°,  20°,  or  100°,  we  require 

7F=  606.5  +  0.305  x    10  =  609.55  heat  units. 
^=606.5  +  0.305  X    20  =  613.60     " 
W^  606.5  +  0.305  x  100  =  637 

For  the  total  heat  of  other  liquids,  Regnault  found  for 

Ether W=    9400  +  0.45000^  -  0.00055556fl 

Acetic  acid W^  140.50  +  0.36644^  -  0.000516^^. 

Chloroform 7F  =    67.00  +  0.1375^. 

Chloride  of  carbon W^    52.00  +  0.14625^  -  0.000172^1 

Bisulphide  of  carbon  .  .  W=    90.00  +  0.14601^  -  0.0004123^1 

If  we  subtract  the  "heat  of  the  liquid"  from  the  total  heat, 
we  have  not  only  the  heat  necessary  for  the  transformation  of 
the  water  into  steam,  but  also  that  required  for  the  outer  work. 
This  heat,  therefore,  we  call  the  ""total  heat  of  vaporization,'^  and 
denote  it  by  r. 

Accordingly  we  have 

r  =  IF  -  (7 (Y.) 

or  putting  for  W  and  q  their  values, 

r  =  606.50  -  0.695^  -  0.00002^2  _  0.0000003^  .    (VI.) 

From  this  formula  we  see  that  the  "total  heat  of  vaporiza- 
tion "  diminishes  as  the  temperature  increases ;  is  less,  for 
example,  at  200°  than  for  100°. 

The  following  formulae  give  the  total  heat  of  vaporization  for 
other  liquids: 


INNER  AND  OUTER  HEAT  OF  VAPORIZATION.         379 

For 

Ether r  =    9400  -  0.07901^  -  0.0008514^1 

Acetic  acid r  =  14.0.5    -  0.13999^  -  0.0009125^1 

Cliloroform r  =    67.00  -  0.09485^  -  0.0000507^'. 

Chloride  of  carbon r=    52.00  -  0.05173^  -  0.0002526^^. 

Bisulphide  of  carbon.  .  .r  =    90.00  -  0.08922^  -  0.0004938^'. 

Instead  of  the  above  complicated  formulae  for  the  vaporiza- 
tion heat  of  water,  Clausius  gives 

r  =  607 -0.708^ (VII.) 

According  to  this,  the  total  heat  of  vaporization  for  100°, 
150",  and  200^  is 

r  =  607  -  0.708  x  100  =  607  -    70.8  =  536.2  heat  units, 
r  =  607  -  0.708  x  150  =  607  -    96.2  =  500.8      " 
r  =  607  -  0.708  x  200  =  607  -  141.6  =  465.4      " 

These  vary  somewhat  from  the  values  found  from  the  more 
exact  formula. 

The  facts  cited  seem  to  indicate  that  liquids  are  converted 
into  steam  sooner,  the  higher  the  temperature  at  which  evapo- 
ration takes  place,  or,  what  is  the  same  thing,  the  greater  the 
pressure  under  which  the  steam  is  generated.  The  reason  is 
evident,  for  the  more  heat  the  liquid  contains,  the  further  apart 
are  the  molecules,  and  the  less  is  the  mutual  attraction  between 
them.  The  "heat  of  the  liquid"  is  evidently  the  so-called 
"sensible  heat."  The  "total  heat  of  vaporization"  is  the  so- 
called  "latent  heat." 

Thus  far  reach  the  extended  and  famous  experiments  of  Eeg- 
nault,  to  whom  our  science  is  greatly  indebted. 

We  pass  now  to  that  which  the  mechanical  theory  of  heat 
deduces  from  these  experiments. 

Inner  and  Outer  Heat  of  Vaporisation. — Heat  of  the  Steam. — 
Of  the  total  heat  imparted  when  1  kilogram  of  water  at  O""  is 
converted  into  saturated  steam  at  any  given  temperature,  a 
portion  goes  to  increase  the  temperature  of  the  water.  This 
we  have  called  the  "heat  of  the  liquid,"  or  the  sensible  heat. 
This  is  the  heat  which  takes  effect  as  vibration  work  of  the 


380  THERMODYNAMICS. 

particles,  and  exists  in  the  water  as  heat.  Another  portion  goes 
to  change  the  water  into  steam.  This  we  call  the  "  total  heat 
of  vaporization "  or  the  latent  heat."^  Of  this  last  portion  a 
part  is  required  for  the  change  of  state  of  the  liquid,  that  is, 
for  the  disgregation  work,  and  another  part  is  required  for  the 
outer  work.  The  first  part  we  call  the  "inner  vaporization 
heat,"  or  the  inne?^  latent  heat,  and  the  second  part  is  the  "  outer 
vaporization  heat,"  or  the  oider  latent  heat.  The  total  heat  im- 
parted, then,  during  the  whole  process  of  the  conversion  of  one 
kilogram  of  water  into  steam,  consists  of  three  parts — the 
sensible  heat  of  the  liquid  at  the  boiling  point,  the  inner  latent 
heat,  and  the  outer  latent  heat.  The  two  last,  taken  together, 
comprise  the  total  latent  heat,  and  the  two  first,  the  "  heat  of  the 
steam,"  since  it  is  the  heat  which  remains  after  subtracting  from 
the  total  heat  imparted  the  heat  required  for  the  performance 
of  the  outer  work.  The  rest  remains  in  the  steam,  part  as 
heat  or  actual  energy  of  motion,  in  the  shape  of  vibration 
work  or  sensible  heat,  and  part  as  energy  of  position,  or  poten- 
tial energy,  in  the  shape  of  disgregation  work,  or  change  of 
state,  or  "  latent  heat." 

Since  the  pressures  are  known  under  which  water  is  vapor- 
ized at  different  temperatures,  and  since,  as  we  shall  see,  we 
can  calculate  the  steam  volume  produced  from  one  kilogram  of 
steam  at  different  temperatures,  we  can  easily  find  for  every 


*  ["Latent  heat  "  has  become  part  of  the  vocabulary  of  our  subject,  and  cannot  now  well  be 
gotten  rid  of,  but  the  term  is  ol)jectionable  unless  properly  understood.  "It  should  be  remem- 
bered that  heat  is  a  kind  of  actual  or  kinetic  energy,  consisting  in  the  invisible  ?«o/ic>ns  of  the  par- 
ticles of  a  body,  and  hence  that  heat  is  not  potential  energy  ;  for  its  ability  to  perform  work 
depends  only  upon  the  heat  motions  of  the  particles,  and  not  on  their  relative  jiositions'^  Heat, 
therefore,  which  is  expended  in  performing  outer  work,  ceases  entirely  to  exist  in  the  bod}-  as 
heat  at  all.  The  term  "  latent  heat "  has  come  down  to  ns  from  a  time  when  heat  was  supposed 
to  be  a  substance,  and  consequently  indestruntihle,  and  although  we  now  know  that  heat,  as  such, 
can  he  put  out  of  existence  by  transformation  into  other  forms  of  energy  which  do  not  affect 
the  thermometer,  still  the  term  remains,  and  continues  to  lead  many  to  believe  that  the  heat 
which  has  been  absorbed  or  disappeared  in  doing  work  still  lurks  concealed  somewhere  in  the 
working  substance  as  heat.  This  is  less  often  the  case  when  external  work  only  is  considered, 
for  the  equivalent  of  the  disappearing  heat  is  then  seen  in  visible. external  work  :  but  when  the 
internal  work  done  in  separating  the  molecules  and  changing  then-  arrangement  is  under  con- 
sideration, the  beginner  is  apt  to  thmk  that  this  kind  of  work  "  don't  count,"  and  that  the  heat 
which  has  been  expended  in  accomplishing  it  still  exi.sts  in  the  body  as  heat,  instead  of  recogniz- 
ing that  in  this  case  also  it  has  disappeared  by  being  transformed  into  another  kind  of  energy— 
that  of  position— which  also  does  not  affect  the  thermometer. 

No  error  can  arise  from  the  use  of  the  term  "  latent  heat,"  however,  if  understood  as  defined 
by  Maxwell :  "Latent  heat  is  the  quantity  of  heat  which  must  be  communicated  to  a  body  in  a 
given  state  in  order  to  convert  it  into  another  state  without  clianging  its  temperature."  (See 
article  by  J.  F.  Klein,  Journal  of  Franklin  Inst.,  April,  1879.)] 


mJYEE  LATENT  HEAT.  381 

temperature  tlie  outer  latent  lieat,  or  "  outer  heat  of  vaporiza- 
tion." 

Then  by  subtracting  this  outer  latent  heat  from  the  total 
latent  heat  we  obtain  the  inner  latent  heat  which  goes  to  trans- 
form the  water  into  steam,  or  to  perform  disgregation  work. 
This  inner  latent  heat  we  denote  by  the  Greek  letter  p,  while 
the  total  latent  heat  is,  as  before,  denoted  by  r. 

As  we  have  seen  in  Chapter  XIV.,  page  361,  at  the  tempera- 
'ture  of  100°  the  inner  latent  heat  is  496.3,  the  outer  latent  heat, 
page  362,  is  40.2,  and  the  sensible  heat  is  100  heat  units  when 
one  kilogram  of  water  at  0°  is  converted  into  saturated  steam 
at  100°. 

Zeuner  has  found  that  the  inner  latent  heat  is  given  by  the 
formula 

p  =  a  —  bt  —  d^, 

where  the  coefficients  a,  h,  c  have  special  values  for  each  dif- 
ferent liquid,  and  t  is  the  temperature  for  which  the  inner  latent 
heat  is  required.     For  water  we  have 

a  =  575.40,         h  =  0.791,        c  =  0,         hence 

P  =  575.40  -  0.791^    ....     (YIIL) 


What  is  the  inner  latent  heat  for  saturated  steam  at  130°  and  at  150^  ? 
For  130^  we  have 

p  -  575.40  -  0.791  x  130  =  4^2.57. 

For  150°, 

p  =  575.40  -  0.791  x  150  =  456.75. 

Zeuner  has  found,  also, 

For 

Ether p^    86.54  -  0.10648f  -  0.0007160/1 

Acetic  acid p  =  131.63  -  0.20184^  -  0.0006280^1 

Chloroform p=    62.44  -  0.11282/  -  0.0000140.1 

Chloride  of  carbon . . . .  p  =    48.57  -  0.06844/  -  0.0002080/1 
Bisuli^hide  of  carbon.  .p=    82.79  -  0.11446/  -  0.0004020/1 


382 


THERMOD  TNAMIGS. 


A 


I 


B 


We  can  now  easily  deduce  a  general  expression  for  the  outer 
latent  lieat. 

Let,  again,  ABCD,  Fig.  63,  be  a  liollow  cylinder.  Upon  the 
bottom  lies  one  kilogram  of  water  at  0°,  and  the  piston  KK 
rests  upon  the  surface.  Upon  the  piston  we  have  the  press- 
ure p.  As  soon  as  the  water  is  heated  to  t",  let  steam  for- 
mation commence.  Let  ti  be  the  height  to 
which  the  piston  is  forced  when  all  the  water 
is  just  evaporated.  Then  the  work  performed 
-j^       by  the  steam  is 

2JU  meter-kilograms. 

Since  a  mechanical  work  of  1  meter-kilogram 
corresj)onds  to  A=^il^  heat  units,  this  work 
represents 

Apu  heat  units. 


This  is,  therefore,  the  general  expression  for 
the  outer  latent  heat.  Since,  now,  the  total 
latent  heat  (r)  consists  of  the  inner  (/o)  and  the 
outer  {Apu),  we  have 


r  —  p  +  AjM 


(IX.) 


C  From  Table  IL,  therefore,  we  can  easily  find 

i»  63.  ^^  £^^  different  pressures  and  temperatures.     We 

have  only  to  add  the  values  of  p  and  Ajm.     Thus,  for  example, 

for  a  pressure  of  1.5  atmospheres,  therefore  for  a  temperature 

of  11L7^  r  =  487.01  +  4L16  =  528.17  heat  units. 

Under  the  assumption  that  Ajni  can  be  calculated  for  every 
pressure  and  temperature,  and  that  r  is  given  by  observation, 
we  have 

p  z=r  —  Apu, 

and  from  this  the  values  of  p  in  the  table  are  calculated. 

Following  Zeuner,  we  have  called  that  work  which  heat 
causes  in  a  body  when  it  raises  its  temperature,  and  changes 
the  aggregation  of  the  molecules,  "  inner  work ; "  and  that  work 
necessary  to  overcome  the  outer  pressure,  "  outer  work."  In 
the  formation  of  steam  we  can  call  that  heat  which  raises  the 
temperature  of  the  water  to  the  boiling  point,  and  converts  it 


HEAT  OF  THE  STEAM.  383 

tlien  into  steam,  the  lieat  of  tlie  steam  or  "  steam  heat.''  This 
we  denote  by  J.  The  "steam  heat"  consists,  therefore,  of  the 
sensible  heat  {q)  and  the  inner  latent  heat  {p),  and  hence 

J=q^-  P   .     .    ■ (X.) 

Thus  we  can  find  J  from  Table  II.,  for  different  pressures. 
Thus,  for  2  atmospheres,  the  steam  heat  J  is  121.42  +  480.00 
=  601.42  heat  units. 

We  can  also  obtain  the  steam  heat  by  subtracting  the  outer 
latent  heat  {Apu)  from  the  total  heat  imparted  (  W).  Hence  we 
have  also 

J=  W-Apu (XL) 

This  expression  evidently  follows,  from  the  preceding,  when 
we  substitute  r  —  Apu  for  p.     Thus, 

J  ^=1  q  -]-  r  —  Apu, 

and  since  q  +  r  =  W 

J=  TV- Apu. 

We  give  below  a  scheme  of  the  manner  in  which  the  heat 
imparted  is  divided  up,  together  with  the  notation  employed. 
The  student  should  make  himself  thoroughly  familiar  with  the 
exact  significance  of  each  letter,  and  their  mutual  relations. 

TOTAL  HEAT. 

TV 

* 

Heat  of  liquid  Inner  latent  Outer  latent 

(sensible  heat).  heat.  heat. 

q  p  Apu 

steam  heat.  Total  latent 

J  heat. 


384  THERMODYNAMICS. 


QUESTIONS  FOR  EXAMINATION. 

Define  specific  heat.  Is  the  specific  heat  of  water  constant  ?  What  two  separate  qnestions 
are  involved  in  the  preceding  ?  Does  the  coefficient  of  expansion  of  water  vary  at  different  tem- 
peratures ?  Why  can  we  neglect  the  disgrei;ation  work  ?  If,  then,  the  disgregation  work  and 
variations  of  outer  work  can  be  disregarded,  to  what  must  any  variation  in  specific  heat  of 
water  be  ascribed  ?  Describe  Regnault's  apparatus  for  investigating  this  question.  Why  must 
the  heat  appearing  in  the  calorimeter  be  precisely  that  received  by  the  water  in  the  boiler  ? 
What  is  Regnault's  experimental  formula  for  tlie  specific  heat  of  water  between  ^,  and  /"  ? 
What  does  it  reduce  to  between  0°  and  Z,"  ?  How  did  he  determine  the  value  of  the  coefficients  ? 
What  is  meant  by  the  "  mean  specific  heat  ?  "  What  is  the  "  heat  of  the  liquid  ?  "  What  letter 
in  our  notation  denotes  it  ?  What  formula  gives  it  ?  Is  the  "  heat  of  the  liquid  "  always  greater 
numerically  than  the  temperature?  What  is  "actual  specific  heat?"  How  does  it  differ  from 
"mean?" 

What  do  you  understand  by  "  total  heat  ?  "  Describe  Eegnault's  apparatus  for  determining 
it.  What  is  the  empirical  formula  for  the  total  heat  ?  What  letter  in  our  notation  denotes  it  ? 
If  you  subtract  the  heat  of  the  liquid  from  the  total  heat,  what  remains  ?  Define  total  heat  of 
vaporization.  What  letter  denotes  it  in  our  notation  ?  What  is  the  formula  of  Clausius  for  total 
heat  of  vaporization  ?  Does  it  give  exact  results  ?  Whtt  do  you  understand  by  sensible  heat? 
What  by  latent  heat  ?  What  is  inner  latent  heat  ?  Outer  latent  heat  ?  If  the  totallatent  heat 
and  the  outer  latent  heat  are  given,  how  can  you  find  the  inner  latent  heat?  What  effect  does 
this  heat  perform  ?  What  letter  in  our  notation  denotes  it  ?  What  is  Zeuner's  formula  for  it  ? 
What  does  the  expression  Apu  denote  ?  Give  the  exact  significance  of  each  letter.  What  does 
r  denote  ?  p  ?  What  is  the  relation  between  r,  p,  and  Apu  ?  What  do  you  understand  by  steam 
heat  ?  What  letter  denotes  it  ?  What  is  the  relation  between  ./,  q,  and  p  ?  Construct  scheme 
which  shows  the  manner  in  which  the  total  heat  IF  is  divided  up.  What  is  the  relation  between 
J,  W,  and  Apu  ?    Give  the  exact  significance  of  each  letter. 


CHAPTEE  XVI. 

CALCULATION  OF  SPECIFIC  STEAM  VOLUME. — EMPIRICAL  FORMULiE  FOR 
THE  INNER  AND  OUTER  LATENT  HEAT,  AS  ALSO  FOR  THE  DENSITY 
OF  STEAM. 

Calculation  of  Sijecif.c  Steam  Volume. — The  question  now  arises, 
How  can  we  calculate,  from  the  temperature  and  pressure  of  the 
steam,  the  outer  latent  heat  {Apu),  or,  what  amounts  to  the 
same  thing,  how  can  we  find  the  specific  steam  volume  ?  As 
soon  as  we  know  this  we  can  easily  determine  the  outer  and 
inner  latent  heat.  At  first  the  specific  volume  was  found  by  the 
combined  law  of  Mariotte  and  Gay-Lussac. 

It  was  assumed,  therefore,  that  saturated  steam  behaved  pre- 
cisely like  a  permanent  gas,*  and  hence  that  its  specific  volume 
could  be  easily  found  from  its  temperature  and  tension.  Thus 
iip  is  the  expansive  force  of  a  permanent  gas,  T  the  absolute 
temperature,  and  v  the  specific  volume,  we  have,  as  seen  already 
in  Part  I., 

2W  =:RT, 

where  B  =  29.272  for  air.     We  have,  therefore, 

_RT 

~    P  ' 

Gay-Lussac  concluded  from  his  experiments  that  the  volume 
of  steam,  at  the  same  temperature  and  tension,  was  always  1.6064 
times  as  great  as  that  of  air.     Hence  for  steam  we  should  have 

V  =  1.6064  X  29.272  -  =47.023  -. 

P P 

*  Since  now  all  the  so-called  "permanent  ga?es  "  have  been  liquefied,  the  term  is  only  to  be 
taken  as  meaning  those  gases  removed  so  far  from  their  point  of  liquefaction  that  the  disgre- 
gation  work  is  ver}'  slight,  or  iiull,  and  which,  under  ordinary  pressure  and  temperature,  remain 
approximatelj''  perfect  gases. 

25  385 


386  THERMODYNAMICS. 

From  the  preceding  we  can  find  upon  these  assumptions  the 
pressure  j/:)  for  any  given  value  of  jr(=  273  +  t),  or  for  p  given, 
can  find  the  temperature.  (This,  at  least,  can  be  found  easily 
from  the  formula  of  Magnus.) 

EXAMPLE. 

What  is  the  specific  volume  of  steam,  according  to  the  above  (incorrect) 
formula,  for  the  temperature  of  100'  and  144°  ? 

For  100°  we  have,  from  Table,  p  =  10334  kilograms,  and  !r=  273  +  100  = 

ono 

373".     Hence  v  =  47.023      ',^.  =  1.6928  cubic  meters.     The  volume  for  144°  is 
10334 

47.023 (since  for  144'  the  pressure  is  4  atmosphei-es),  or  0.4749  cubic 

meters. 

The  specific  weight,  or  the  weight  of  1  cubic  meter,  must  evidently,  according 

to  G-ay-Lussac,  be  always  ^  „„„ ,   =  0.622  of  that  of  air  at  the  same  temperature 

and  pressure. 

Later  investigations,  especially  the  calculations  of  the  me- 
chanical theory  of  heat,  have  shown  that  the  experiments  of 
Gay-Lussac  are  not  exact,  and  that  the  conclusions  drawn 
from  them  are  incorrect.  The  determination  of  steam  volume 
based  upon  these  experiments  and  assumptions  cannot,  there- 
fore, lay  claim  to  much  accuracy.  The  exact  determination  is, 
however,  essential  to  any  reliable  theory  of  the  steam  engine, 
and  for  this  reason  all  such  theories  having  such  incorrect  basis 
are  inexact.  To  Clausius  belongs  the  credit  of  being  the  first 
to  show  how  steam  volumes  may  be  found,  by  the  aid  of  the 
mechanical  theory  of  heat,  with  far  greater  accuracy  than  ac- 
cording to  the  earlier  methods.  Before  we  give  his  method, 
we  would  call  attention  to  the  following  customary  terms  and 
notations  of  the  mechanical  heat  theory. 

Customary  Terms  and  Notation  for  Steam. — We  call  the  volume 
occupied  by  1  kilogram  of  water  atO°  the  "specific  ivater  volume," 
and  denote  it  by  d.  The  volume  of  1  kilogram  of  water  at  0^ 
is,  then,  ff  cubic  meters.  Let  us  again  assume  this  specific  water 
volume  inclosed  in  a  cylinder  with  a  movable  piston.  When 
heat  is  imparted  we  have  a  gradual  evaporation  of  the  water. 
Suppose  that  at  any  moment,  of  the  1  kilogram  of  water,  x  kilo- 


TERMS  AND  NOTATION  FOR  STEAM.  387 

grams  are  steam  {x  being  then  less  tlian  1),  then  1  —  x  kilo- 
grams are  still  water.  Since  now  1  kilogram  of  water  occupies 
the  space  g,  if  we  disregard  the  slight  increase  of  bulk  of  the 
water  when  heated,  1  —  x  kilograms  will  occupy  the  space 
il  —  x)  (J  cubic  meters.  "When  the  entire  kilogram  of  water  is 
just  converted  into  steam,  let  its  volume,  that  is,  the  specific 
steam  volume,  be  s.  The  volume  of  x  kilograms  will  then  be 
sx  cubic  meters.  Therefore  the  entire  volume  of  the  1  —  x 
kilograms  of  water,  and  of  the  x  kilograms  of  steam  will  be 

{\  —x)  (J  +  xs  cubic  meters. 

This  volume,  whose  weight  is  still  one  kilogram,  and  which 
consists  partly  of  steam  and  partly  of  liquid,  we  denote  by  v, 
so  that  we  have 

V  =  (1  —  a?)  (T  +  xs, 
or 

r)  =  G  —  xff  +  xs  —  ff  -{-  {s  —  (j)x. 

That  is,  the  specific  volume  of  the  mixture  of  steam  and 
water  is  equal  to  the  specific  water  volume  (c)  plus  the  pro- 
duct of  the  steam  weight  {x)  into  the  difference  of  the  specific 
steam  and  water  volumes.  Clausius  denotes  this  difference 
{s  —  g)  by  u,  so  that 

V  =  c  +  ux. 

The  value  of  g  is  readily  determined  by  experiment  for  dif- 
ferent liquids.     Thus  it  has  been  found 

For 

Water a  =  0.001    cubic  meters. 

Ether c  =  0.0013 

Alcohol 0-  =  0.0013 

Acetic  acid tr  =  0.0012 

Chloroform a-  =  0.0006 

Chloride  of  carbon c  =  0.0006 

Bisulphide  of  carbon . . .  .g  —  0.0008 

Hence  we  see  g  is  so  small  with  regard  to  ux  that  it  may  be 
disregarded,  and  we  have  simply 


388 


THEBMOD  TNAMICS. 


EXAMPLE. 

What  is  the  volume  of  a  quantity  of  steam  and  water  at  100\  whose  weight  is 
1  kilogram,  which  consists  of  0.3  kilograms  steam,  and  0.7  kilograms  water? 

According  to  Table  II.,  for  100'  m  —  1.649  cubic  meters,  hence  ux  =  1.649  x 
0.3  =  0.4947  cubic  meters,  and  6  +  %iz  =  0.001  +  0.4947  =  0.496  cubic  meters. 

Example  2.— One  kilogram  of  steam  and  water  has  a  temperature  of  130.8° > 
of  which  0.5  cubic  meter  is  steam.     How  much  does  it  weigh? 

Smce  at  the  temperature  130.3'  ?<  =  0.647  cubic  meter,  and  6  is  very  small 
compared  to  s,  we  have 

0  499 
0.5  =  0.001  +  0.647Z.     Hence  x  =  ^-^  =  0.77  kilograms. 

The  water  is  therefore  1  —  0.771  =  0.229  kilograms. 


Steam  Volume,  calculated  according  to  the  Principles  of  TJiermo- 
dynamics. — Let  us  now  see  how  the  specific  steam  volume  may 
be  calculated. 

Let  OA,  Fig.  64,  be  the  specific  water  volume  (c).  We  con- 
_.  ceive  it  again  inclosed  in  a 


0 


B 


liiiillllll illlllilllllillliiliiliiililB 


pm.q: 


^6^ 


iifc" 


cylinder   of  1  square  meter, 
cross-section,  and  the  piston 
"n         loaded  with  p  kilograms.     In 
such  case,  then,  c  is  the  dis- 
tance  of  the   surface  of  the 
water,  or  of  the  piston,  from 
bottom  of  the  cylinder.  After 
the  water  is  heated  to  t°,  let 
the  steam  generated  just  have 
the  pressure  p.     Therefore  t^ 
is    the   boiling   point   corre- 
'X  sponding  to  the  pressure  p. 
By  further  addition  of  heat 
^^<^-  ^-  steam  is  formed,  and  the  pis- 

ton will  be  lifted.  When  all  the  water  is  converted  into  steam, 
let  the  specific  steam  volume  he  0  G  =  s.  In  other  words,  the 
j^iston  now  stands  at  the  height  OG  from  the  bottom.  The 
distance  passed  through  by  the  piston  is  then  OG  —  OA  = 
s  —  (T  =  u.     During  this  the  pressure  p  is  constant. 

Let  us  now  suppose  the  sx^ecific  steam  volume  s  to  expand 


AH 


^ 


SPECIFIC  STEAM  VOLUME.  389 

adiabatically  until  tlie  temperatiire  has  sunk  by  the  very  small 
quantity  r,  and  the  pressure  'p  has  become  jj^.  For  this  small 
decrease  of  temperature  we.  may  assume  that  the  saturated 
steam  acts  as  a  permanent  gas,  and  hence  the  line  CD,  which 
is  an  adiabatic,  for  this  very  short  distance,  is  a  straight  line. 

Now  let  the  steam  at  pressure  jh  be  compressed,  and  heat  at 
the  same  time  abstracted,  so  that  the  pressure  2h  remains  con- 
stant until  the  specific  volume  of  water  and  steam  is  OH,  the 
volume  of  the  mixture,  or  the  position  of  H,  being  so  chosen 
that  when  the  remaining  steam  is  compressed  adiabatically 
along  BE,  it  shall  all  be  condensed,  and  come  back  to  its  origi- 
nal temperature  and  its  original  volume, 

"We  have  in  this  way  completed  a  simple  cycle  process,  and 
the  outer  work  performed  is  given  by  the  shaded  area  BCDE, 
or  by  the  product 

(p  -p,)BC=  {p-pi)u. 

The  heat  imparted  from  A  to  G,  or  B  to  C,  is  greater  than 
that  abstracted  from  D  to  E,  and  the  difference^  must  equal  the 
outer  work. 

If  we  denote  the  heat  imparted  by  Q,  the  work  performed  is 
(page  184,  Part  I.)  ^^ 

where  T  =  273  +  Ms  the  absolute  temperature  of  the  steam 
from  Bio  C,  T-r  =  "^1^  +  t-r  that  from  D  to  E.  We  have 
therefore 

{p  -pO  u=-j^r.        LU  ----   -•    ^  ^ 
Denoting,  for  the  sake  of  brevity,  p  —  p)^  by  jt, 


0 .1.      ^^  ^  4f 

AT    7t  1  -T-     i.f. 


Now  Q  is  evidently  that  heat  which  must  be  imparted  to  1 
kilogram  of  water  at  the  temperature  t,  in  order  to  convert  it, 


390  THEBMOD  YNAMICS. 

under  constant  pressure,  into  steam  of  f,  or  it  is  the  total  latent 
lieat  of  vaporization,  r.     Hence 

r       T      4:24:r    r  /vtt  \ 

"  =  zr-i=^-;;  •  •  •  •  (™-) 

This  formula  is  the  most  important  hitherto  deduced  from 
the  application  of  the  mechanical  theory  of  heat  to  steam. 
From  it  and  the  preceding  equations,  those  which  follow  can  be 
easily  derived. 

The  ratio  -  of  the  change  of  temperature  to  the  change  of 

pressure,  can  be  exactly  determined  by  the  formulae  on  page  369 
and  following,  or  by  means  of  the  calculus.  The  fact  that  the 
differences  of  pressure  for  the  temperatures  t  +  1,  f,  and  t  —  1 
are  nearly  the  same,  furnishes  the  means  of  determining  this 
ratio  in  an  elementary  manner  with  all  necessary  exactness. 
Thus,  for  example,  if  we  wish  to  determine  w  for  a  temperature 
of  80°,  and  take  the  difference  of  pressure  (rt)  for  SO''  and  79°, 
this  difference  will  be  relatively  too  small.  If  we  take  the  dif- 
ference of  pressures  for  80°  and  81°,  it  will  be  relatively  too 
large.  If  we  take  the  mean  of  both  differences,  it  will  be  very 
closely  the  increase  or  decrease  of  pressure  for  a  very  small 
change  of  temperature.  If  we  denote,  therefore,  the  pressure 
at  81°  by  p,  and  at  79°  by  pi,  we  have  for  r  about,  say,  -jV  of  ^ 
degree. 

'    •  ^'  —  1  0    •   1  0  2         ' 

or  generally 


TT  '      p  —  Pi        P  —  Pi 

Here  p  and^:)i  denote  the  pressure  in  kilograms  per  square 
meter.     Table  I.  gives  them  in  centimeters  of  barometer.     To 
reduce  those  in  the  table  to  kilograms  per  square  meter  we 
have  to  multiply  the  tabular  values  in  millimeters  by  J-^|f^-. 
We  have  then 

r  _       2  X  760       _  0.147 
TT      10334  {p  —  pi)      p  —  pi' 

where  j9  and ^^i  are  given  in  millimeters  of  mercury  column. 


SPECIFIC  STEAM  VOLUME.  391 

The  formula  for  v,  becomes  then 

_  424  X  0.147r  _  62.328     r  ,^^^^ , 

The  specific  steam  volume  s  we  can  at  once  obtain  from  ti 
by  adding  the  specific  water  volume  ff  =  0.001.     Thus 

G2.328  r       „  „„^       ,  .         ^ 
s  = -=  +  0.001  cubic  meters. 


EXAMPLE. 

What  is  the  value  of  ^i  for  0'  and  for  80'  ? 

According  to  Rontgen's  formula  for  finding  the  pressure  from  the  tempera- 
ture, already  given,  we  have  for  +  1',  p  =  4.91,  and  for  —  1%  p^  =4.25  milli- 
meters. Further,  the  latent  heat  of  vaporization,  r,  for  0'  is  606.5  heat  units. 
Hence 

62.328  606. 


4.91  -  4.25  273 


=  209.78  cubic  meters. 


By  the  aid  of  the  calculus,  and  using  Regnault's  values  for  the  pressures,  we 
should  obtain  m  =  210.66.     The  difference  is  then  very  small. 

Again,  for  81%  we  have  from  the  table,  ^j  =  369.258,  and  for  79%  ^i 
=  340.464  millimeters.     Smce  for  80'  r  -  550.618, 

62.328  550.618        „  ,_       ,. 

=  3.376  cubic  meters. 


869.258-340.464        273  4 

Columns  of  Table  II.  gives  the  accurate  values  of  u  from  0.1  to  14  atmos- 
pheres. 

More  recent  investigations  of  Tate  and  Fairbairn  have  shown 
that  the  specific  steam  volumes  calculated  according  to  thermo- 
dynamic principles  agree  quite  closely  with  the  results  of 
observation,  and  far  better  than  those  found  by  the  combined 
law  of  Mariotte  and  Gay-Lussac. 

If  the  pressures  of  the  other  liquids  had  been  determined 
from  degree  to  degree,  we  could  find  with  the  same  exactness 
the  value  of  u  for  their  vapors  also.  As  this  is  not  the  case,  we 
can  only  find  approximate  values  of  u,  but  values,  nevertheless, 
quite  close  to  the  actual. 


392  THEBM0DTNAMIC8. 

Example  1. — What  is  the  steam  volume  of  1  kilogram  of  ether  when  evapo- 
rated at  40'  ? 

The  pressure  of  such  vapor  at  50'  has  been  found  to  be  1264.83  millimeters, 
and  at  30%  634.80  millimeters.  The  difference  630.03  corresponds  to  20\  The 
difference  for  2°,  that  is,  for  i  +  1  and  t  —  1  degrees,  or^  — />,,  is  63.003  milli- 
meters. Since,  from  the  empirical  formulae  already  given,  r  for  ether  at  40"  is 
89.48,  and  since  T  =z  273  +  40  ==  313,  we  have 


62.328   89.48       ^  ^^^      ,. 
^0T3-'313-  =  ^-^^^'^^^^^^^^^^^- 


Zeuner  finds  by  calculus,  0.285. 

Example  2. — What  is  the  steam  volume  of  alcohol  at  50°  ? 

For  60°  the  pressure  has  been  found  350.21  millimeters,  and  for  40°,  133.69. 
Difference  for  20°,  216.52,  for  2°,  21.652.  For  50%  r  is  233.79,  and  r=323, 
hence 

62.328      233.79      „  „„, 
^  =  2T65-2^-32-3--^-0«4- 


Calculation  of  the  Outer  and  Inner  Latent  Heat. — Since  we  can 
now  find  u  for  every  temperature  and  pressure,  it  is  easy  to 
determine  the  outer  work.  Tlius  if  p  is  tlie  constant  pressure, 
this  outer  work  is  simply  pu  meter-kilograms. 


EXAMPLE. 

What  outer  work  will  be  performed  by  the  steam  generated  from  1  kilogram 

of  water  at  150",  the  constant  pressure  being  equal  to  its  own  tension  ? 

10334  X  3581  23 
For  150°  the  pressure,  from  Table  I.,  is  3581.23  millimeters,  or ^^^ — '- — 

=  48673.14  kilograms  per  square  meter.  We  find  from  Equation  XIII., 
M  =  0.384  cubic  meters,  hence  p^i  =r  48673.14  x  0.384  =  18690.43  meter-kilo- 
grams. For  I,  §,  i  of  a  kilogram  converted  into  steam  we  should  obtain  only  J 
or  J,  etc.,  of  18690.43  meter-kilograms. 


The  heat  required  for  this  work,  which  we  call  the  outer 
latent  heat,  is 


^P'^''  —  tcTa  lieat  units. 
424 


OUTER  AND  INNER  LATENT  HEAT.  393 

Tlierefore  tlie  outer  latent  heat  in  tlie  vaporization  of  1  kilo- 
gram of  water  at  150'^  is 

— TTT^-  =  44.08  heat  units. 
424 

In  this  way  the  outer  latent  heat  has  been  calculated  for  Table 
II.,  column  7,  for  from  0.1  to  14  atmospheres.  We  see  that  it 
increases  with  the  temperature  or  expansive  force  of  the  steam. 

Since,  now,  we  know  the  total  latent  heat  r  from  Regnault's 
experiments  and  empirical  formulae,  and  the  outer  latent  heat 
can  be  calculated  as  above,  we  can  find  the  inner  latent  heat  p 
from  the  equation 

p  —  r  —  Apu. 

It  has  been  found  thus,  and  is  given  in  column  6  of  Table  II. 
It  is  evident  that  it  must  decrease  as  the  temperature  increases, 
since  the  outer  latent  heat  increases. 

Finally,  the  steam  heat  J  can  be  found  from  the  equation 

J=  W-Apu. 

Since  for  water  u  differs  from  s  only  by  about  0.001  cubic 
meters,  and  s  is  very  great  with  reference  to  0.001,  we  can  put 
u  in  place  of  s.  In  such  case,  u  cubic  meters  require  the  inner 
latent  heat  p,  and  1  cubic  meter  requires 

—  heat  units. 


The  expression  -  gives  us,  therefore,  the  heat  units  neces- 
sary for  the  inner  latent  heat  of  1  cubic  meter  of  steam  under 
various  pressures.  Since  we  shall  have  frequent  occasion  to 
make  use  of  it,  it  is  given  in  Table  II.,  column  9. 

Empirical  Formulce  for  the  Calculation  of  the  Inner  and  Outer 
Latent  Heat  and  of  the  Specif  c  Steam  Volume. — For  the  mechani- 
cal engineer,  to  whom  the  easy  and  accurate  determination  of 
w  and  s  is  of  great  importance,  empirical  formulae  are  very  de- 
sirable, so  that  even  withoiit  tables  he  can  find  these  quantities 


394  THERMODYNAMICS. 

for  different  pressures  witli  sufficient  exactness.  The  calcula- 
tion of  u,  already  given,  is  too  involved  for  practical  men.  Zeu- 
ner  was  the  first  to  meet  this  want.  As  we  have  already 
remarked,  he  has  given  for  the  inner  latent  heat  p,  the  simple 
formula 

p  =  575.40  -  0.791^. 

Now  we  have  from  IX. 

Apu  =  r  —  p, 
or,  since  r  =  W  —  q, 

Apu  —  W—  p  —  q. 

For  W  we  have,  according  to  Kegnault,  606.5  +  0.305#,  and 
for  q,  t  +  0.00002^2  +  0.0000003^,  hence 

Apu  =  606.5  +  0.305;5  -  (575.40  -  0.7910 
-  (^  +  0.00002^^  -  0.0000003^3), 

or,  after  reduction, 

Apu  =  31.1  +  0.096^  -  0.00002^2_  o.0000003i-l 

This  expression  enables  us  to  determine  the  outer  latent 
heat  {Apu)  from  the  temperature  alone. 

If  we  divide  the  last  equation  by  Ap,  we  have 

31.1  +  0.096^  -  0.00002^2  -  0.0000003^^      ,^„7  ^ 

u^ ^^,        .     (XIV.) 

Since  A  is  known,  we  only  need  to  know  the  variation  of  p 
with  the  temperature  in  order  to  find  u.  For  this  we  can  either 
make  use  of  Table  I.  or  II.,  or  lacking  these,  of  some  one  of  the 
expansion  formulse  already  given. 


EXAMPLE. 

What  is  the  difference  between  the  specific  steam  and  •water  volume,  or  what 
is  the  value  of  u,  for  130"  and  for  200'  ? 

According  to  Table  I.,  the  pressure  at  130^  is  2030.28mm.,  hence  the  pressure 

in  kilograms  per  square  meter,  p,  is  ^^^^  +  2030.28  ^  27602  kil.     Now  t  --=  130, 

t-  =  16900,  f  ^  2197000. 


INNER  AND   OUTER  LATENT  HEAT.  395 

Hence 

424  (31.1  +  0.096  x  130  -  0.00003  x  16900  -  0.0000003  x  219700) 
"  "  27602  ' 

_  424  (31.1  +  12.48  -  0.338  -  0.6591) 
""  -  27602  ' 


424  (43.58  -  0.9971)  _  424  x  42.583   .  ._  .. 
27602       -    27602    "  ^'^^^^^ 


For  200'  the  pressure  is  11688.96mm.  =  158937  kil.  t  -^  200,  f  =  40000, 
t^  =  8000000,  and 

_  424  (31.1  +  0.096  x  200  -  0.00002  x  40000  -  0.0000003  x  8000000), 
^~  158937  ' 

_  424(31.1  +  19.2-0.8-24)  _  424(50.3-3.2), 
~        158937        ~    158937    ' 

_29970^_ 

~ 158937  ~ 


In  tlie  lack  of  tables,  we  may  use  for  the  calculation  of  p,  for 
temperatures  less  than  100°,  the  formula  of  Magnus.  Above 
100^,  that  of  Kontgen. 

We  have  also  the  following  empirical  formulae  given  by  Zeu- 
ner  for  the  other  liquids  already  named  : 

Ether Apu  =  7.46  +  0.02747^  -  0.0001354^1 

Acetic  acid Apu  =  8.87  +  0.06185^  -  0.0002845^1 

Chloroform .Apu  =  4.56  +  0.01797^  -  0,0000367^^. 

Chloride  of  carbon Apu  =  3.43  +  0.01671^  -  0.0000546^1 

Bichloride  of  carbon Aidu  =  7.21  +  0.02524^  -  0.0000918^^. 


EXAMPLE. 

What  volume  of  steam  is  generated  from  1  kilogram  of  ether  when  evaporated 
at  40"  ? 

The  pressure  at  40°   is  907.04mm.,  or  ^-^^  x  907.04  =  12297.5  kilograms. 

t  =  40,  t^  =  1600,  hence 

A  X  12297.5  X  tt  =  7.46  +  0.02747  x  40  -  0.0001354  x  1600, 


396  THEBMODTNAMIGS. 

or 

1        -.oon^K  Qo^oo  8.3422x424 

424  "  1^^^^-^  ^  ^  =  ^•^^^^'  ^^         ''  =        12297        ' 

or 

u  =  0.287  cubic  meters. 

On  page  392  we  found  for  u  0  283  cubic  meters,  or  but  little  different  from 
the  empirical  formulae. 

Density  of  Saturated  Steam. — The  preceding  is  sufficient  to 
show  that  the  view  of  Gay-Lussac,  that  the  density  of  saturated 
steam  is  always  0.6225  of  that  of  air,  is  not  correct. 

Since  the  specific  steam  volume,  that  is,  the  volume  of  1  kilo- 
gram of  steam  s  =  ti  +  o;  the  weight  y  of  one  cubic  meter  of 
steam,  which  we  call  the  "  specific  tveight,"  will  be 

11  1  Ti 

y=  -  =i = TTT^i^  kilograms. 

^       s     u  +  0      tc  +  0.001         ^ 

Thus,  for  example,  for  150",  the  weight  of  one  cubic  meter  of 
steam  is 

^  ^         2.597  kil. 


'       0.384  +  0.001  ~  0.385 

The  value  of  y  is  given  in  column  11  of  Table  II.  for  from 
0.1  to  14  atmospheres.     For  air  we  have  always 

therefore  the  specific  volume  is 

29.272  r 

v= , 

P 
and  the  specific  weight 

1  n 

For  example,  for  150°  the  pressure  of  steam  is  4.71  atmos- 
pheres, hence  the  pressure  p  in  kilograms  per  square  meter  is 
4.71x10334  =  48673  kilograms.      Since    now    r=  273 +  150 


DENSITY  OF  SATURATED  STEAM.  397 

=  423,  for  this  temperature  and  pressure,  the  weight  of  one 
cubic  meter  of  air  would  be 

_        48673        _  48673  _ 
^^  ~  29.272  X  423  ~  12382  ~  "^-^"^^  ^^^• 

On  the  other  hand,  the  weight  y  of  one  cubic  meter  of  satu- 
rated steam,  under  the  same  conditions,  is  really  2.597  kilo- 
grams, as  already  computed.  Accordingly,  the  density  6  of  the 
steam,  that  is,  the  ratio  of  its  weight  to  that  of  an  equal  amount 
of  air  under  the  same  conditions,  is 

r     2.597  _ 

^  =  ^  =  3":930  -  ^•^^^- 

If  we  calculate  in  this  way  the  specific  weights  of  steam  and 
air  for  different  temperatures  and  pressures,  we  find  for  the 
density, 

For 

0.1  0.5  1  2  5         10  atmospheres 

d  =  0.621      0.633      0.640      0.648      0.662      0.676, 

from  which  we  see  that  the  density  increases  tolerably  rapidly 
with  increasing  temperature.     Hence 

cannot  be  a  constant  quantity  as  with  gases  is  the  case.  Satu- 
rated steam  follows  some  other  law  than  this.* 

[*  The  relation  pv  =  ^T  therefore  holds  good  only  for  those  gases  so  far  removed  from  their 
point  of  saturation  that  they  may  be  considered  as  perfect.  Zeuner  has  recently  shown  that  for 
steam,  whether  saturated  or  superheated, 

k-\ 

pv  =  BT-  Cp  ^ 

in  which  £  =  '^'  ~  ,  and  Cp  =  0.4805,  and  k  =  1.333,  hence  B  =  50.933,  and  C=  192.50,  p  being 
in  kilograms,  or  if  p  is  in  atmospheres,  then  for  both  saturated  and  superheated  steam 
pv  =  BT-  C^JP  ,  where  B  =  0.0049287,  and  C  -  0.187815.    See  Appendix  to  Chap.  XXIII.] 


398  THERMODTNAMICS. 


QUESTIONS   FOR   EXAMINATION. 

Define  specific  steam  volume.  How  was  this  first  calculated?  Upon  what  assumption  was 
this  calculation  founded  ?  Give  the  relation  between  pressure,  volume,  and  temperature  for  a 
gas.  What  does  R  stand  for  ?  What  did  Gay-Lussac  conclude  from  his  experiments  ?  Was 
this  conclusion  correct  ?  Is  the  old  method  of  calculation  of  specific  volume  correct  ?  Why 
not  ?    How  does  saturated  steam  differ  from  a  perfect  gas  ? 

Define  what  is  meant  by  specific  water  volume?  What  letter  denotes  it  in  our  notation  ? 
What  is  specific  steam  volume  ?  What  letter  denotes  it  ?  What  does  u  denote  in  our  notation  ? 
What  relation  exists  between  v,  cr,  and  u,  if  x  is  the  weight  of  steam  in  one  kilogram  of  steam  and 
water  ?    What  does  v  denote  ?    o-  ?    ?/  ? 


Deduce  the  expression  u  =  --^,  .  —.    What  is  the  exact  simificancc  of  each  of  the  letters  ? 

AT  TT 

Is  this  formula  Important  ?    Why  ?    Show  how  to  find  in  an  elementary  manner  the  approxi- 
mate value  of  the  ratio  -  in  any  given  case.    Explain  now  the  use  of  Tables  I.  and  II. 

If  u  is  given,  how  can  you  find  the  outer  work  ?    The  outer  latent  heat  ?    How  can  you  find 
the  inner  latent  heat  ?    How  can  we  find  the  total  latent  heat  ?    What  is  the  steam  heat  ?    What 

does  -  denote  ?    What  is  Zeuner's  formula  for  the  inner  latent  heat  ?    How  can  you  find  from 

it  the  outer  latent  heat  and  the  specific  steam  volume  ? 

\Yhsit  is  specific  weight  ?    What  letter  in  our  notation  denotes  it  ?    What  is  meant  by  density 
of  steam  ?    How  does  this  vary  for  different  temperatures  and  pressures  ? 


CHAPTEE  XYII. 

CUEVE    OF    CONSTANT    STEMI    WEIGHT. — EMPIRICAL    FORMULA. — DE- 
PORTMENT  OP   STEAM  WHEN  IT  EXPANDS   PERFORMING  WORK. 


Curve  of  Constant  Steam  Weight. — If  we  lay  off  the  volumes 
of,  say,  1  kilogram  of  steam  for  successive  pressures,  as  abscis- 
sas, and  tlie  corresponding 
pressures  as  ordinates,  we 
obtain  a  curve  (Fig.  65) 
which  represents  the  law 
according  to  which  the 
volume  changes  with  the 
pressure.  We  may  call 
this  the  ^^  curve  of  constant 
steam  iceight."  For  a  press 
ure  of  1  atmosphere  the 
volume  of  1  kilogram  of 
steam  is,  from  Table  II., 
1.649  +  0.001  =  1.650  cub. 
m.  Taking  only  2  decimal 
places,  we  have  s  —  1.65. 
Hence  OA  =  1.65  units  to 
any  given  scale,  and  the  perpendicular  AB  \&  laid  off  according 
to  another  given  scale, 
atmospheres  is  about  0, 
and  CD  —  2,  and  so  on. 

"We  see  that  the  volumes  decrease  nearly  inversely  as  the 
pressures,  that  is,  that  the  volumes  are  2,  3,  4  times  less  when 
the  pressures  are  2,  3,  4  times  greater.  If  this  were  accurately 
the  case,  the  relation  between  the  pressures  and  the  specific 
volumes  would  be 


The  volume  of  1  kilogram  of  steam  at  2 
cubic  meters.     Therefore,  0(7=0.86, 


jps  —  ip^Sx  —'Pih,  etc. 


399 


400  THERMODYNAMICS. 

As,  however,  is  seen  from  tlie  Figure,  this  is  not  exactly  tlie 
case.  Zeuner  has  found  by  calculation  that  the  law  for  the 
curve  of  constant  steam  weight  is  given  very  closely  by  the 
formula 

ps-'-'^' =^ iW"^' =  PiSi"^' '    .     •     .     (XV.) 

where  the  exponent  of  s,  s^ ,  etc.,  is  only  0.0646  greater  than  in 
the  preceding  formula  which  gives  the  law  of  Mariotte  for  per- 
manent gases. 

Evidently  these  products,  since  they  are  all  equal  to  each 
other,  must  equal  a  constant  value,  and  this  value  is,  according 
to  Zeuner,  1.704,  so  that 

1.104:  =^  2^s'-'^' =  Pi^i''^' '    •     •     •     (XYI.) 


EXAMPLE. 

What  is,  according  to  this  forrmila,  the  specific  volume  of  saturated  steam  at 
a  pressure  of  two  atmospheres  ? 

o-        V,  o        1,  '""VrTOi       i.o64fi.p^3^        ,  log  0.854 

Since  here^  ==  2,  we  have  s=    y  — —  =r      y  0.854,  or  log  s  =     °  = 

From  this  formula  we  obtain  s  as  well  as  u  with  great  exactness.     It  is  also 
more  convenient  for  calculation  than  Equation  XIV.     "We  may  also  obtain  from 

it  the  specific  weight  y  of  the  steam.     Thus  r  —  —  and  s  =  —,  hence 

s  y 


1.704=,,  (i)' 


^'■""^i^y^o.Bsesp, 


1.0B46, 


;^  =^1.0646  X    ■  yO.5868,    or  finally 

;k  =  0.6061  x^"-9393. 

By  means  of  this  formula  Zeuner  has  found  the  specific 
weight  of  steam  for  different  pressures,  and  compared  with  the 
values  obtained  by  previous  calculations.  The  coincidence  is 
so  great  that  only  occasionally  is  there  a  deviation  of  one  unit 


CURVE  OF  SATURATION.— CBITICAL   TEMPERATURE.     401 

in  the  tliird  decimal  place.     Hence  the  last  formula  is  of  great 
practical  use. 

We  have  assumed  above  that  the  volumes  and  pressures  for 
one  kilogram  of  steam  are  taken  as  abscissas  and  ordinates. 
Instead  of  this,  we  might  have  taken  the  volumes  of  one-half, 
one-third,  etc.,  kilogram  of  steam,  together  with  the  correspond- 
ing pressures,  and  thus  obtained  a  curve.  This  new  curve 
would  have  the  same  law  as  the  above,  and  will  only  differ  in 
having,  with  reference  to-  the  same  axes,  a  different  beginning 
and  end.     It  is  represented  in  the  Figure  by  the  dotted  line 


• 

s»                            ^nc2        it*T~ 

\                                      ^^                ^ 

Is.                                     -.c.  ^ 

s.^=^^-^^:rr-.r^ 

Pig.  66. 


[Curve  of  Saturation. — Critical  Temperature. — Let  a  series  of  isothermals  be 
drawn,  &?,  A^B^S-^^T^,  AoBiS-iTi.  etc.,  as  m  the  following  Figui-e,  of  which  the 
portions  A^B^,  A^B.,  repre- 
sent the  changes  of  pressure 
and  volume  of  the  fluid  at  con- 
stant temperature  in  the  liquid 
state  ;  B^S„  B.,S.,,  etc.,  the 
process  of  eA^aporation,  and 
S^T,,  S.,To,  etc.,  the  expan- 
sion of  the  superheated  vapor 
at  constant  temperature.  A 
curve  drawn  through  the  points 
Su  S.2,  etc.,  will  represent  the 
changes  which  maybe  under- 
gone by  the  fluid  while  it  re- 
mains entirely  in  the  state  of 
saturated  vapor.  It  is,  there- 
fore, called  the  curve  of  satura- 
tion. The  volume  of  all  fluids  in  the  state  of  saturated  vapor  decreases  as  the 
pressure  and  temperature  increase,  and  thus  the  curve  of  saturation  slopes  down- 
ward from  left  to  right,  as  shown  in  the  Figure. 

On  the  other  hand,  the  volume  of  every  liquid  at  the  boiling  point  increases 
with  pressure  and  temperature.  Therefore  a  curve  drawn  through  the  series  of 
points  B^,  B.,,  etc.,  will  slope  in  the  opposite  direction  to  the  curve  of  saturation, 
and  the  two  will  approach  each  other  as  the  pressure  increases,  and  at  length 
meet.  The  physical  interpretation  of  this  is  that  at  a  certain  temperature  the 
liquid  and  gaseous  states  become  continuous,  there  being  no  marked  separation, 
such  as  that  observed  in  the  ordinary  processes  of  evaporation  and  condensation, 
between  them.  This  is  called  the  critical  temperature  of  the  fluid.  Above  this 
temperature  the  fluid  retains  the  properties  of  a  gas  under  any  pressure  however 
great. 

It  is  supposed  that  the  so-called  permanent  gases  resist  condensation  into  the 
liquid  form  so  greatly  because  the  lowest  temperatures  which  we  are  able  to  pro- 
duce ordinarily  are  still  above  their  critical  temperatures. 

For  certain  substances  the  critical  temperature  has  been  accurately  deter- 
26 


402 


THEBMOD  YNAMIC8. 


mined.     For  instance,  that  of  carbonic  acid  is  at  87.7"  Fahr.,  or  31.17'  C,  and 
"tlie  corresponding  pressure  of  saturation  is  about  74  atmospheres. 

There  are  a  few  substances,  however,  which  can  readily  be  brought  to  the 
critical  temperature.  Water  reaches  it  at  about  720.6'  Fahr.,  or  382.55'  C.  The 
corresponding  pressure  of  saturation  has  not  been  determined.  But  both  tem- 
perature and  pressure  are  far  higher  than  those  met  with  in  the  practical  applica- 
tions of  steam.] 


Deportment  of  Steam  ivJien  it  Expands  Performing  WorJc. — A 
knowledge  of  tlie  deportment  of  saturated  steam,  when  it 
expands  while  performing  work,  is  of  especial  importance  in 
practice,  as,  by  means  of  it,  we  are  in  a  position  to  estimate  more 
exactly  tlie  action  of  steam  in  tlie  steam  engine.  It  was  formerly 
assumed  tliat  steam  in  expanding  not  only  remained  saturated, 
but  also  tliat  the  steam  weight  did  not  change ;  that,  therefore, 
the  expansion  took  place  along  the  curve  of  constant  steam 
weight.  Pambour  especially,  to  whom  we  owe  the  first  com- 
plete and  systematic  theory  of  the  steam  engine,  assumed  this 
principle  in  his  development,  and  after  him  all  writers  down  to 
recent  times  accepted  it  as  correct.  Although  now  the  saturated 
steam,  under  the  given  conditions,  remains  saturated,  as  is  indi- 
cated by  the  older  Observations  of  Pambour,  and  the  more 
recent  observations  of  Hirn,  still  the  steam  iveight  is  not  constant, 
in  other  words,  expansion  does  not  take  place  according  to  the 
curve  of  constant  steam  weight.  This  fact  can  only  be  made 
apparent  by  the  aid  of  the  mechanical  theory  of  heat,  as  was 
done  in  1851,  almost  simultaneously,  by 
Clausius  and  Eankine.  It  is  easy  to  show 
that  the  expansion  of  steam  in  a  steam 
engine  does  not  follow  the  curve  of  con- 
stant steam  weight. 

Let  OA,  Fig.  67,  be  the  steam  volume 
behind  the  piston,  its  temperature  t 
=  144°,  and  pressure  p  =  4  atmospheres 
=  41336  kilograms  per  square  meter. 
Let  the  volume  OA  of  1  kilogram  be 
0.447  cubic  meters.  If  this  steam  ex- 
pands to  the  volume  OD  —  0.507  cubic 
^^^-  ^"  meters,  the  temperature  sinks  to  140.44° 

and  the  pressure  to  CD  =3.5  atmospheres  =  36169  kilograms 
per  square  meter.     If  now  the  steam  during  this  expansion 


EXPANSION  OF  SATURATED  STEAM.  403 

remains  saturated,  and  tlie  steam  weight  constant,  tlie  curve 
BC  i^  a  portion  of  the  curve  of  constant  steam  weight,  and  the 
shaded  area  ABCD  denotes  the  mechanical  work  performed 
during  expansion.  The  contents  of  this  area  are,  considering 
it  as  a  trapezoid, 

^^  +  ^^  X  AD, 


2 

41336  +  36169 


(0.507  -  0.447)  =  2325 


2 
meter-kilograms.     This  work  represents 

-t'^t-  —  5.49  heat  units. 
424 

Now  the  steam  heat  at  4  atmospheres  of  1  kilogram  of 
steam  is 

e/=  145.31  +  461.50  =  606.81  heat  units, 

and  at  8.5  atmospheres 

J=  140.44  +  465.26  =  605.70  heat  units. 

The  difference,  1.11  heat  units,  is  not  sufficient  to  perform 
the  work  of  2325  meter-kilograms.  For  this  purpose  4.38  heat 
units  more  are  necessary. 

Since  now,  according  to  our  assumption,  no  heat  is  imparted 
from  without,  we  must  conclude  that  the  steam  condenses,  and 
that  the  condensation  supplies  the  lack  of  heat  of  4.38  heat 
units. 

Since  steam  condenses  during  the  expansion,  the  work  done 
cannot  be  so  great  as  when  the  steam  weight  is  constant,  and 
hence  the  curve  of  expansion  must  approach  the  axis  OX  more 
rapidly  than  EC,  which  is  a  portion  of  the  curve  of  constant 
steam  weight.  If,  then,  the  end  pressure  is  the  same,  the  end 
volume  cannot  be  OD,  but  must  be  less  than  OD.  If  OF  is 
this  volume,  and  FE  =  CD  the  final  pressure,  the  work  during 


404 


THEBMOB  YNAMIG8. 


expansion  is  given  by  ABEF.  The  volume  OF  and  tJie  press- 
ure EF  must  correspond  to  tlie  volume  and  pressure  of  the 
remaining  saturated  steam.  The  point  E  must  therefore  lie 
in  a  curve  of  constant  steam  weight,  where  the  weight  is  less 
than  1  kilogram.  Since  the  expansion  takes  place  without 
heat  being  added  from* without,  the  curve  BE  must  be  part  of 
an  adiabatic. 

Our  example  shows  plainly  that  during  expansion  steam  is 
condensed,  or  else  that  heat  must  be  imparted,  but  it  does  not 
give  the  exact  value  of  this  heat,  since  the  work  during  expan- 
sion is  not  given  by  ABCD  but  by  the  less  area  ABEF.  If, 
therefore,  we  wish  to  find  this  heat  we  must  adopt  another 
method.  This  has  been  done  by  Clausius,  in  his  "  AhJiandlung 
ilber  llecJiamsche  Warmetheorie,"  1864,  and  we  shall  now  pro- 
ceed to  point  it  out. 

Sup230se  in  a  prismatic  vessel  a  mixture  of  steam  and  water  of 
the  temperature  t  and  pressure 
AB,  Fig.   68.     Of  this  mixture 
let  31  kilograms  be  liquid  and 
B  P5i!iiniiiii!wiii^  '>n  kilograms  steam.     Upon  the 

steam   jaresses   a  piston  whose 
pressure  is  AB. 

"We  impart  heat  to  the  water 
while  assuming  the  pressure  re- 
mains constant.    In  this  case  all 
the  heat  goes  to  form  steam,  and 
AS  ID  llir  ^^    therefore    latent.      Suppose 

X    that  thus  mi  kilograms  of  water 
^^'  ^  are  vaporized,  so  that  we  now 

have  in  all  m  +  Wi  kilograms  of  steam.  If  now  r  denotes  the 
latent  heat  when  1  kilogram  of  water  at  the  temperature  t,  and 
under  the  constant  pressure  AB,  is  evaporated,  the  heat  im- 
parted is 

miTi  heat  units (1). 


Now  let  the  entire  steam  volume  m  +  m^  expand  adiabati- 
cally.  The  expansion  is  then  at  the  expense  of  the  heat  of  the 
mixture,  and  the  temperature  sinks.  If  we  suppose  the  expan- 
sion CG  to  be  very  small,  the  decrease  of  temperature  is  slight. 
Denote  it  by  r,  so  that  at  G  the  temperature  of  the  mixture  is 


,       EXPANSION.   OF  SATURATED  STEAM.  405 

t  —  T.  It  is  clear  that  CG  is  a  portion  of  an  adiabatic.  We 
may  sii23pose  now,  in  opposition  to  our  calculations,  that  from 
C  to  G,  ih  kilograms  of  steam  are  formed. 

Now  we  compress  tlie  steam,  assuming  that  it  is  always 
saturated,  from  the  volume  OH,  temperature  t  —  t,  and  press- 
ure HG,  so  that  pressure  and  temperature  remain  constant. 
We  must  then  abstract  heat  during  compression.  This  com- 
pression is  carried  to  a  point  F,  so  chosen  that  when  from 
there  on  the  steam  is  compressed  adiabatically  the  mixture 
will  retake  its  original  condition,  and  we  shall  have  again  M 
kilograms  of  water,  and  m  of  steam  at  the  temperature  t.  Then 
the  temperature  from  F  to  B  has  been  increased  by  r,  and  the 
work  which  the  steam  performed  by  expansion  from  C  to  G  has 
been  received  again  from  F  to  B.  We  have  thus  here  a  simple 
cycle  process.     Inner  work  has  been  neither  gained  nor  lost. 

Let  now  r.^  be  the  heat  which  must  be  abstracted  from  1  kilo- 
gram of  steam  at  the  temperature  t  —  T,m  order  to  obtain  1 
kilogram  of  water  at  the  temperature  t  —  t,  then  if  from  G  to 
F,  m-2  kilograms  are  condensed,  the  heat  abstracted  is 

m^Ti,  heat  units (2). 

The  excess  of  the  heat  imparted  over  that  abstracted  is 

m-ii\  —  m-iV^  heat  units (3). 

By  this  excess  a  certain  mechanical  work  is  obtained,  repre- 
sented by  the  area  BCGF,  which,  since  CG  and  BF  are  very 
small,  we  may  regard  as  a  parallelogram.     The  area  is  then 

BC  X  {AB  -  FE). 

If  we  denote  the  difference  AB  —  FE  by  tt,  we  have 

BG  X    7T. 

If  now  the  volume  of  1  kilogram  of  steam  at  t°  is  u,  that  of 
mi  kilograms  is  miU.     Hence  BC  =  riiiU,  and 

BC  x  7t  —  in{ii7t. 

The  heat  corresponding  to  this  work  is 

Ami^iTT .....     o     .     .     (4). 


406  THE R  MOD  TNA  MIC 8. 

This  heat  must  be  equal  to  (3),  hence 

m{t\  —  m^r^  =  Am{U7r (5). 

Now  the  steam  weight  generated  on  the  path  BCG  is 

mi  +  Ui, 

and  that  condensed  on  the  path  GFB  is 

mo  +  n2 , 

assuming  that  during  the  compression  BF,  n^  kilograms  are  de- 
posited. Since  at  the  end  of  the  process  we  have  the  original 
quantity  of  water  and  steam, 

mi  +  oil  =  m-i  +  712 (6). 

Hence 

m2  =  mi +  011  —  712 (7). 

Substituting  this  value  in  (5), 

miO'i  —  ouiv,  —  HiT-i  +  7iir-i  —  AwiiUrt   .     .     .     (8). 

We  can  eliminate  iii  and  n-,  from  this  equation  as  follows: 
We   have    assumed  during  the  expansion  CG  steam  to   be 
formed,  therefore  heat  amounting  to 

7ii7^2  heat  units 

taken  from  the  existing  water  and  steam  in  order  to  form  the 
Hi  kilograms.  If  the  specific  heat  of  the  water  is  c,  then  the 
heat  abstracted  from  the  water  31  —  mi  is 

(if—  TOi)cr  heat  units. 

But  heat  is  also  taken  from  the  existing  steam  mass  m  +  mi. 
If  we  suppose  that  1  kilogram  of  saturated  steam  at  f  must 
give  U23  h  heat  units  in  order  to  remain  saturated  at  ^  —  1  de- 
grees,* then  the  m  +  mi  kilograms  of  steam  lose 

(m  +  mi)  hr  heat  units. 

*  [We  see  therefore  that  h  plays  the  part  of  a  specific  heat.  Wc  may  consider  it  as  the 
"  specific  heat  of  saturated  steam  for  constant  steam  quantity/  J 


EXPANSION  OF  SATURATED   STEAM.  407 

(We  shall  see  liereafter  that  //  is  negative,  so  that  during  ex- 
pansion the  steam  does  not  lose  heat,  but  gains  it,  as  should 
be  the  case.) 

We  have  then 

n-^r,  =  {31—  m^cT  +  (m  +  rri])  lir     .     .     .     (9). 

From  F  io  B  the  steam  is  compressed  adiabatically.  If,  now, 
on  the  way  CG  heat  is  abstracted  from  the  existing  mass,  or 
the  way  BF  it  is  given  back.  Since,  by  supposition,  n^  kilo- 
grams are  condensed,  the  heat  set  free  is 

n^r~^  heat  units. 

This  is  divided  among  M  kilograms  of  water,  and  m  of  steam. 
The  first  accordingly  receives 

il/cr  heat  units, 
and  the  second 

*  mlir  heat  units. 
Hence 

ly^r-i  =  Met  +  mlir (10). 

Substituting  (9)  and  (10)  in  (8) 
m{i\  —  iiixTi  —  (If—  m^cr  —  (m  +  m^  In  +  Mcr  +  mlit  —  Amiun^ 

or  reducing 

ri  —  i\  -{-  cr  —  JiT  =  AuTt    ....     (11). 

Now  the  total  heat  of  1  kilogram  of  steam  of  the  temperature 
t  is  606.5  +  0.305^  heat  units.  Or  if  the  specific  heat  of  the 
water  from  which  the  steam  is  generated  is  c,  and  the  latent 
heat  ri, 

606.5  +  0.305^  =  ct+r,     .     .     .     .     (12). 

For  1  kilogram  of  steam  of  the  temperature  #  —  r  we  have,  in 
like  manner, 

606.5  +  0.305  {t-r)  =  r.,  +  c{t-j)     .     .     (13). 

Subtracting,  we  have 

0.305r  =  ri  -  i\  +  ct. 


408  THERMODYNAMICS. 

Substituting  in  (11) 

0.305 r  -  hr  =  Au7t (14). 

From  Equation  (XII.),  page  390,  we  have 


T  ^  =  Au7r (15). 


and  from  this  and  (14)  we  have 


A  =  0.305-1^ (XVII.) 

or 

Since  -=j  =  ^r=^ — -  is  always  greater  than  0.305,  h  is  negative. 

We  see,  therefore,  that  ivhen  saturated  steam  expands  performing 
work,  so  that  the  temperature  sinks,  loe  have  not  to  abstract  hut  to  add 
heat  in  order  to  keep  it  saturated.  And  if  saturated  steam  is  com- 
pressed, heat  must  not  he  added  hut  abstracted  in  order  to  keep  it 
saturated.  Otherwise  the  steam  is  superheated,  and  has  a  higher 
pressure  than  saturated  steam  of  the  same  volume.  The  heat 
imparted  in  the  first  case,  and  abstracted  in  the  second,  is  for 
1  kilogram,  for  a  rise  or  fall  of  1  degree, 

'  =  '■'''- Tlhrf 

As  r  —  W—  q,  or 

r  =  606.5  -  0.695^  -  0.00002f^  -  0.0000003^ 

7      A  QA^       606.5  -  0.695^  -  0.00002^  -  0.0000003^ 
h  =  0.305 -— 273  +  ^ 

Since,  according  to  Clausius,  we  have  with  good  exactness 
r  =  607  -  0.708^, 


SPECIFIC  HEAT  OF  STEAM  FOB  CONSTANT  WEIGHT.      409 
we  may  also  write 


or  finally, 


Aid  +   t 


k  =  lM3-^^    .    .    .     (XYHL) 


EXAMPLE. 

How  many  heat  units  must  be  imparted  to  1  kilogram  of  saturated  steam  at 
100%  when  it  expands  performing  work,  and  the  temperature  sinks  1'  ? 

Prom  Table  II.  we  have,  since  r  =  p  +  Apu,  for  100%  499.30  +  40.20  =  523.50 
heat  units,  hence 

h  =  0.305  -  5#^^  =  0.305  -  ^H^  =  0.305  -  1.438  =  -  1.133. 
273+100  372 

Tlie  problem  wliicli  we  have  discussed  can  be  solved  in  a 
simpler  manner.  , 

Suppose  the  cycle  process  completed.  Tlie  work  performed 
is  given  by  tlie  shaded  area  BCDE.  Suppose  now  that  the  ex- 
pansion on  the  path  CD  had  extended  until  the  temperature 
had  sunk  V  instead  of  r.    Then  the  work  L  would  be 

i=-iy[r-(r-i)]=^ 

where  r  is  the  latent  heat  from  B  to  C.  We  know  that  this 
work  can  only  be  gained  when  the  heat  imparted  along  BC  is 
greater  than  that  abstracted  along  GF,  because  the  work  which 
is  performed  by  expansion  CG,  by  reason  of  inner  heat  of  the 
steam,  is  equal  to  that  which  is  expended  in  the  compression 
FB. 

The  work  performed  expressed  in  heat  units  is  therefore 

r 


Q  =  AL  =  ~ 


Hence  the  volume  OG  oi  the  saturated  steam  of  t  degrees 
contains  ^  lieat  units  more  than  the  volume  OF,  of  the  satu- 
rated steam  of  ^  —  1  degrees.     Now,  1  kilogram  of  saturated 


410  THEBMOD  YNA  MICS. 

steam  of  f  has  only  0.305  heat  units  more  than  1  kilogram  of 

^  —  1  degrees.     Since  ^  >  0.305,  we  have  to  impart  during  ex- 

T 

pansion  y^  —  0.305  heat  units,  or  to  abstract  during  compres- 
sion BE,  the  same  amount,  in  order  to  keep  the  steam  saturated. 
This  is,  therefore,  the  heat  added  or  abstracted,  which  we  have 
denoted  by  A,  so  that 

h=^  -  0.305. 

Or,  if  we  consider  the  heat  imparted  during  expansion  as 
negative, , 

h  =  0.305  -  ^ . 

Our  formula  shows  that  h  is  variable  with  the  temperature. 
We  see,  especially  from  our  equation,  page  409,  that  h  is  greater, 

that  is,  is  nearer  zero,  the  greater  t,  since  the  quotient  ^„q 

Alo  -\-t 

diminishes  with  increasing  temperature. 

Heat  Imparted  or  Abstracted  for  Great  Differences  of  Tempera- 
ture.— In  the  following  tabulation  we  have  given  the  heat  neces- 
sary in  order  to  keep  1  kilogram  of  steam  of  10,  20,  30,  to  120 
degrees,  saturated  and  uncondensed  during  its  expansion  and 
cooling  of  one  degree.  The  same  heat  is  requisite  to  keep  the 
same  quantities  of  steam  saturated  at  the  same  temperatures, 
when  the  temperature  is  raised  1  degree,  but  the  heat  must  be 
then  abstracted. 

Temperature      0  10  20  30  40  50  GO 

Value  oih     -  l.OlT     -  1.814     -  1.718     -  1.628     -  1.544     -  1.465     -  1.391 

Temperature         70  80  90  100  110  120 

Value  of  A         -1.321        -1.255        -1.192        -1.183        -1.077       -1.024. 

If,  now,  it  is  required  to  determine,  for  example,  what  heat 
must  be  imparted,  in  order  that  1  kilogram  of  saturated  steam 
at  100'  may  expand  gradually  to  1  kilogram  of  saturated  steam 
at  0^,  we  must  determine  the  mean  of  A  between  0^  and  lOO'', 


SPECIFIC  HEAT  FOR  CONSTANT  STEAM  WEIGHT.       411 

and  multiply  by  tlie  number  of  terms  by  whicli  the  mean  was 
determined.     We  may  find  the  mean  best  by  Simpson's  rule. 

If  we  have  a  number  of  quantities  occurring  at  equal  inter- 
vals, and  denote  them  by  P^,,  Pj,  P.. .  .  .  P^^i,  ^o  that  n  is  the 
number  of  intervals,  the  mean  is 

P  =  [IP,  +  P,  +  P,  +  P  ,^-  .  .  .  P„  _,  +  IP„)  ^  n. 

If  the  number  of  intervals  n  is  even,  viz.,  2,  i,  6,  8,  etc,  the 
rule  gives  for  the  mean 

P=  (Po  +  4P,  +  2P2  +  4P3+  .  .  .  4P„_i  +  P„)^Zn. 

If  we  wish,  then,  to  find  the  mean  of  h  betw^een  0""  and  100°, 
according  to  the  first  formula,  we  must  put  for  Pq  1.917,  for 
Pi  1.814,  for  Po  1.718,  finally,  for  P„  1.133.  Then  -|Po  =  0.958, 
and  IP,,  =  0.567,  hence 

P  =  (0.958  +  1.814  +  1.718  +  1.628  +  1.544  +  1.465  +  1.391  + 
1.321  +  1.255  +  1.192  +  0.567)  --  10  =  14.853  ^  10  =  1.485 ;  or 
since  h  is  generally  negative,  A  =  —  1.485. 

If,  then,  1  kilogram  of  saturated  steam  expands,  performing 
work,  from  100^  to  0",  and  still  remains  saturated  and  uncon- 
densed,  we  must  impart  on  the  average,  for  each  degree  of 
cooling,  1.485  heat  units.     The  entire  heat  imparted  is  then 

Q  =  1.485  X  100  =  148.5  heat  units. 

In  the  same  way  we  may  find  for  1  kilogram  of  steam  whose 
temperature  sinks  during  expansion  from  80^  to  0°, 

Q  =  1.558  X  80  =  124.64  heat  units. 

Zeuner  has  given  a  table,  which  gives  the  amount  of  heat 
which  must  be  imparted  when  1  kilogram  of  saturated  steam 
of  1,  2,  3,  etc.,  atmospheres  cools  by  expansion  to  0°  and  remains 
all  steam.     In  Table  III.  we  have  given  these  values  of  Q,  as 

7' 

well  as  the  corresponding  values  of  -^  . 


412  THERM0DTNAMIG8. 


EXAMPLE.   • 

How  many  units  of  heat  must  be  imparted  to  1  kilogram  of  saturated  steam 
of  5  atmospheres,  when  it  expands  in  the  cylinder  of  an  engine,  performing  work, 
down  to  1  atmosphere,  and  yet  still  remains  saturated  and  uneondensed  ? 

From  Table  III.,  for  a  pressure  of  5  atmospheres. .  Q    =  200.46 
And  for  1  atmosphere Qi  =  148.47 

The  heat  imparted  from  5  to  1  atmos.  is  then 51.99  heat  units. 

This  amoimt  of  heat  is  too  great  to  be  supplied  by  the  hot  cylinder  sides,  as 
has  been  assumed  by  the  followers  of  Pambour.  If  therefore  no  heat  is  imparted 
from  without,  so  much  steam  must  be  condensed  as  will  furnish  the  necessary 
heat.  This  amount  of  steam  can  indeed  be  relatively  very  small,  since  the  latent 
heat  of  steam  is  great  with  respect  to  the  heat  required. 


Deportment  of  other  Vapors  during  Expansion. — The  formula 
for  li  was 

A  =  0.305-^. 


In  this  0.305  is  tlie  amount  of  lieat  wliicli  one  kilogram  of 
steam  of  ^  +  1  degrees  possesses  more  than  1  kilogram  of  t  de- 
grees, because  1  kilogram  of  ^  +  1  degrees  has 

606.5  +  0.305  {t  +  1)  heat  units, 

and  one  of  t  degrees  has 

606.5  +  0.305^  heat  units, 

and  the  difference  is  0.305  heat  units. 

The  total  heat  of  1  kilogram  of  ether  steam  of  ^  +  1  degrees 
is,  as  we  have  given  it, 

94  f  0.45  (^  +  1)  -  0.00055556  {t  +  1)^ 
or 

94.450556  +  0.4489^  -  0.000556^2, 


DEPOBTMENT  OF  OTHER   VAPORS.  413 

For  t°  it  is 

94  +  0.45^  -  0.000556^1 

Tlie  difference  is 

0.450556  -  O.OOllf. 

Hence  tlie  expression  for  7i  for  etlier  steam  is 


h  =  0.450556  -  0.0011^  -  ^ . 
Since,  however  (page  378),  r  =  94  -  0.079^  -  0.00085/^  we  have 
h  =  0.450556  -  0.0011^  -  ^  "  ^'^^^^  '  '-'^'^'^ 


273 +  t 

hT  =  (273  +  t)  0.450556  -  0.0011^  (273  +  0 

-  94  +  0.079^  +  0.00085^^    or 

hT=  29.003  +  0.229^5  -  0.00025^1 

If  we  find  the  formula  for  water  steam  in  a  similar  manner, 
we  have 

hT=  -  523.23  +  t  +  0.00002^^  +  0.0000003^^. 

We  see  from  this  formula,  that  even  for  very  high  tempera- 
tures h  is  still  negative,  as  we  have  already  concluded  from  the 
form  of  other  formulae.  We  see  from  the  formula  for  ether 
steam  that  h  is  positive  even  when  the  temperature  is  very  great. 
This  kind  of  steam  therefore,  must  have  heat  extracted  from  it 
during  expansion,  if  no  part  of  it  is  condensed.  This  peculiar 
deportment  of  ether  steam  was  first  pointed  out  by  Hirn.  All 
other  vapors  which  we  have  named,  in  fact,  all  for  which 
Regnault  has  determined  the  sensible  heat  and  latent  heat  of 
vaporization,  comport  themselves  like  water  steam,  and  for 
them  h  is  therefore  negative. 


414  THERMODYNAMICS. 


QUESTIONS   FOR  EXAMINATION. 

What  is  the  curve  of  constant  steam  weight '  When  saturated  steam  expands,  performing 
woik.  does  it  remain  saturated  ''  Is  the  steam  weight  constant '!  If  not.  can  you  prove  that  it 
is  not  ?  If  saturated  steam  is  compressed,  and  heat  at  the  same  time  abstracted  so  that  tlie  tem- 
perature is  kcut  constant,  what  talies  place?  If  no  heat  is  abstracted  y  Itu  expands  ])erform- 
iii!.'^  work,  and  heat  is  not  added  '  How  many  iieat  units  must  be  imparted  to  1  kilogram  of 
saturated  steam  at  100'^  to  kceii  il  saturated  and  tincondensed,  when  it  expands  i)erformingwork, 
till  the  temperature  is  'JO"  ■/    fcjiippose  no  heai  is  added,  how  much  sleam  will  be  condensed  ? 


CHAPTER  XYIII. 

HEAT   CURVES   OF    STEAM  AND   LIQUID   MIXTURES. — CONSTRUCTION   OF 
THE   SAME. — TECHNICAL   APPLICATIONS. 

A.  Isothermal  Curve. 


Form  of  the  Curve. — The  isothermal  curve  has  been  defined  as 
that  which  gives  the  change  of  condition  of  a  body  when  the 
temperature  is  kept  constant.     For  gases  this  was  a  curved  line 
which  made  aj)parent  the   law 
of  Mariotte.       Now    we    know 
that  if  for  saturated  steam  the 
pressure  is  constant,  the   tem-                      g 
perature  is  constant  also.      If, 
then,   AB  =  p  (Fig.  69)   is  the 
pressure  of  the  steam  in  a  mix- 
ture of  steam    and  water,  this 
pressure   remains    constant   so 
long  as  the  temperature  is  the 
same.     Heat  added  to  the  wa- 
ter simply   vaporizes   some   of       ij j- D"""^ 

it,  the    volume    increases,   and  ^     ^„ 

'  .  Fig   69. 

the  isothermal  for  the  mixture 

is  a  straigU  line  parallel  to  OX*  Since  for  a  higher  or  lower 
temperature  the  pressure  p  is  greater  or  less,  the  line  5(7  will 
be  at  a  greater  or  less  distance  from  OX. 


Outer  and  Inner  WorJc  during  Expansion. — ^Let  OA  =  vhe  the 


*  [Here  therefore  the  isothermal  and  isopiestic  lines  coincide.] 

415 


41 6  THERMOD  YNA  MICS. 

initial  specific  volume  of  tlie  mixture  (volume  of  1  kilogram), 
then,  as  already  proved,  page  387, 

where  it  can  be  taken  from  Table  II.  for  given  pressure  and 
temperature.  If  now  we  heat  the  mixture  till  the  volume  is  v^, 
we  have 

Since  the  ordinate  AB  —  iJ  describes  the  rectangle  ABCD, 
the  outer  work  performed  is 

L  —i:)  (vi  —  v)  =  ipu  {xx  —  X) ; 
hence 

_Vx  —  V 


In  practice  v^  —  v  is  generally  given,  w  can  be  taken  from 
Table  II.,  and  thus  the  weight  of  water  vaporized  {x-^  —  x)  can 
be  found. 

Now  what  is  the  entire  amount  of  heat  imparted  ?  All  this 
heat,  as  we  know,  goes  to  vaporize  the  water.  Of  this  the 
outer  latent  heat  is 

AL  —  Ap  (^1  —  V)-—  Apu  {xi  -  x), 
while  the  inner  latent  heat  is 

•         P{x^-x) (XX.) 

The  total  amount  of  heat  is  then 


Q^r{x,-x)  =  r'^ ^    .     .     .     (XXI.) 


Q  =  {Apu  +  p)  {x,  -  x) 


(Apu  +  p)  ^^^-^ .     .     .     .     (XXII.) 


The  value  of  Aj)u  +  p,  as  also  u,  can  be  found  from  Table  IL 


ISOTHERMAL  CURVE  FOR  STEAM.  417 

Example  1. — The  cylinder  of  a  non-condensing  engine,  working  with  full 
pressure,  has  a  cross-section  of  0.174  square  meter,  and  a  stroke  of  1.048  meters. 
The  steam  pressure  j9  is  3i  atmospheres,  and  the  number  of  revolutions  per  min- 
ute is  24,  What  is  the  theoretical  work  per  second,  and  how  much  heat  is  re- 
qiured  ? 

The  steam  quantity  per  stroke  is  0.174  x  1.048  =  0,182  cubic  meter.     Hence 


V,  -u=  0.182, 

and  the  work 

per 

stroke 

is 

L-- 

=^(i'i  ■ 

-I')  =  10334  X  3^  X  0.182  = 

=  4702  m. 

kil. 

The  work 

per 

second 

is 

?-^^  X  4702  =  3761  meter-kilograms. 

or 

3761       ^^ , 
„^     =  50  horse  power. 

For  the  heat  required  per  stroke, 

Q  =  {Apu  +  p) 


Since  Apxi  +  p  for  3i  atmospheres  is,  from  Table  II.,  =  465.26  +  43.27  = 
508.53,  and  «  =  0.507,  we  have 

Q  =  508.53  "^  =  182.56  heat  units. 

Hence  the  heat  per  second  is 

182.56  X  U  =  146.05  heat  units. 

If  all  this  heat  had  been  converted  into  outer  work,  we  should  have  had 
424  X  146.05  =  61925  meter-kilograms,  while  in  reality  we  have  only  3761,  or 
hardly  the  16th  part.  Now,  perhaps,  only  half  the  heat  of  the  fuel  acts  to 
vaporize  the  water,  so  that  we  utilize  only  the  32d  part  of  the  heat  of  the  fuel. 
Finally,  even  this  is  but  the  total  work  of  the  engine,  and  from  it  we  must  sub- 
tract the  prejudicial  resistances,  in  order  to  find  the  useful  work.  Since  these 
resistances  take  about  50  per  cent,  from  the  total  work,  we  have  only  ,j-4-th  of  the 
heat  of  the  fuel  actually  utilized. 

Example  2. — A  condensing  engine  sends  0.182  cubic  meter  of  steam,  at  a 
27 


418  THEBMODYNAMICS. 

pressure  of  -|-\;th  of  an  atmosphere,  into  the  condenser,  where  the  pressure  is  con- 
stant.    What  work  is  necessary,  and  how  much  heat  is  taken  from  the  steam  ? 
The  work  required  is 

10344  X  -,V  X  0.182  =  188.08  meter-kilograms. 

From  Table  II.,  Apu  +  p  for  -^,  atmosphere  is  538.85  +  35.46  =  574.31,  and 
u  =  14.55.    Hence 

0  182 
Q  =  574.81  ^^  =  7.179  heat  units. 

14.00 


B.  IsoDYNAMic  Curve. 

The  isodynamic  curve  gives  tlie  law  of  cliange  of  p  and  v, 
when  the  inner  work  is  constant. 


Equation  and  Construction  of  the  Curve. — Suppose,  as  before, 
one  kilogram  of  mixture  to  consist  of  x  kilograms  of  steam  and 
1  —  X  oi  water.  The  sensible  heat  of  the  mixture  is  q,  and 
hence  the  steam  heat  is 

q  +  xp, 
and  the  inner  work  is 

-j{q  +  xp). 

If  now  we  have,  after  adding  heat,  a?i  kilograms  of  steam  and 
1  —  ail  of  water,  and  the  sensible  heat  q^,  and  the  inner  latent 
heat  Pi,  we  have  for  the  "  steam  heat "  in  the  new  state 

gi  +  «iPi, 

and  the  inner  work 

-J  {qi  +  x,p,). 


For  the  isodynamic  curve  then 


—  (q  +  xp)=j-{qi  +  a!iPi), 


q  +  xp  =  qi  +  x^pi  ....     (XXni) 


ISODYNAMIC  CURVE  FOR  SATURATED  STEAM. 


419 


Tills  is  tlie  equation  of  the  isodynamic  curve.  In  order  to 
construct  the  curve  we  must  know  the  abscissa  and  ordinate 
for  different  points.  If  we  assume  p  and  x  known  for  the  ini- 
tial condition,  then  from  p  we  know  t,  u,  q,  and  p.  The  corre- 
sponding volume  is  given  by 

V  —  xu  +  (T     or     V  =  xu. 

If  now,  px  is  given  for  a  second  position,  we  know  at  once  $i 
and  /9i,  and  since  we  know  g  +  xp  for  the  initial  condition,  we 
have 

q  \-xp  —  qx 


x^ 


and  then  from 


Vi  =  0:^1^1  +  pi 


can  find  v^  —  for  the  second  condition. 

Thus  let  p  —  6  atmospheres  and  x 
from  Table  II., 

(^=153.74,     p=  454.99,     and 
u  =  0.363. 

Hence 

V  =  a;2i  +  0-  =  0.8  X  0.363  +  0.001 
=  0.291  cubic  meter. 

Lay  off  now  OA  =  i}  =  0.29,  Fig. 
70,  and  AB  =  5.  Then  5  is  a 
point  of  the  isodynamic  curve. 
We  can  now  calculate  v^  for  2h 
—  4  atmospheres.  For  this,  qx 
=  145.31,  A  =  461.5,  and  ic^  =  0.447, 
hence 

^^_q+xp-qx 
Pi 
153.74  +  0.8  X  454.99  -  145.31 


0.80  kilograms,  then 


461.5 

=  0.807  kilogram.  ^'*'-  ™" 

Therefore  0.007  kilogram  of  water  are  vaporized.    For  Vx  we  have 
Vx  =  XxUx  +  0-  =  0.807  X  0.447  +  0.001  =  0.362  cubic  meter. 


420  THERMODYNAMICS. 

Lay  off  00  =  0.362  and  CB  —  4,  and  D  is  a  second  point  on 
tlie  curve.  In  the  same  way  we  can  determine  the  volumes  for 
pressures  of  3  atmospheres,  2,  and  1  atmosphere,  as  shown  in 
Fig.  70. 

The  curve  joining  all  the  points  thus  found  is  the  isodynamic 
curve  for  a  mixture  of  steam  and  water. 

We  see  here  also,  that  as  in  the  curve  of  constant  steam 
weight,  the  volumes  increase  as  the  pressures  decrease. 

The  curve  can  be  represented  then  by  an  equation  of  the 
form 

2w''—piVy''  =  pz^'T,  etc. 

Zeuner  found  that  when  x  is  originally  —  1  kilogram,  and 
then  the  steam  compressed  according  to  the  isodynamic  curve, 
n  =  1.0456.  For  the  curve  of  constant  steam  weight,  n  =  1.0646. 
The  curve  of  constant  steam  weight  approaches  the  axis  of  X 
more  rapidly  therefore  than  the  isodynamic  curve,  and  lies 
therefore  between  the  latter  and  the  isothermal. 

From  the  preceding  we  see,  that  during  expansion  of  steam 
along  the  isodynamic  curve,  water  is  vaporized,  and  during  com- 
pression, is  condensed.  Thus,  as  we  have  seen,  for  a  pressure 
of  4  atmospheres,  x^  =  0.807  kilogram,  while  for  5  atmospheres, 
X  was  0.8,  and  for  3  atmospheres,  0.815,  etc. 

Outer  Work. — Heat  Required. — In  order  to  determine  the  work 
performed  during  expansion,  we  determine  the  area  of  ACDB, 
then  of  CDFE,  etc.,  considering  them  as  trapezoids.  Thus  for 
example,  for  the  outer  work  during  expansion  from  ^  =  5  to 
^1=  4  atmospheres,  we  have 

L  =  ^-tPl  (^,^_  ^)  =  10334  X  4.5  (0.362-0.291)  =  3302  met.-kil. 

Since  further,  the  inner  work  is  constant,  all  the  heat  im- 
parted goes  to  outer  work.     This  heat  is  then 

Q  =  AL  =  ^ljX  3302  =  7.79  heat  units. 

This  curve  is  of  little  value  in  practice,  hence  we  will  not 
discuss  it  further. 


ADIABATIC  CURVE  FOB  8ATUBATED  STEAM. 


421 


C.    Adiabatic  Cueve. 

This  curve  gives  tlie  law  of  variation  of  p  and  v,  wlien  no 
heat  is  either  imparted  or  abstracted  during  the  change  of  con- 
dition. 


Equation  and   Construction  of  the   Curve.— To  construct  this 
curve  we  must  find  from  a  given  pressure  or  temperature  the 
corresponding  volume.      In  this  connection  we  refer  to  what 
has  been  said  in  the  Appendix 
to    this    chapter,    and   advise 
that  it  be  read  before  the  fol- 
lowing : 

Let  OA,  Fig.  71,  be  the 
volume  of  1  kilogram  of  water 
at  0°.  If  this  water  is  not 
partly  vaporized  when  heat  is 
imparted  to  it,  it  must  be 
loaded  with  a  certain  weight 
or  subjected  to  a  certain 
pressure.  Call  this  pressure 
AB.  Suppose  now  the  tem- 
perature of  this  water  is 
raised  gradually  to  1,  2,  3,  etc.,  degrees.  To  prevent  vapori- 
zation the  pressure  AB  must  be  correspondingly  increased. 
When  the  temperature  of  the  water  is  100'',  the  pressure  is 
10334  kilograms.  The  imparting  of  heat  and  increase  of  press- 
ure is  thus  conducted  in  the  same  manner  as  for  permanent 
gases  in  the  Appendix.  We  have  then  here  a  certain  "  heat 
weight."  If  c  is  the  mean  specific  heat  of  water  between  0  and 
t  degrees,  the  heat  weight  imparted  for  this  rise  of  tempera- 
ture is 


0  A 


c   ,  ,    273  +  ^ 

A  1°S  ^"*  "273- 


^lognat    2^. 


If  we  denote  this  by  ^ ,  we  have 


r  =  c  log  nat  ^  =  2.3026  c  log  -^  .     (XXIY.; 


422  THERMODYNAMICS. 

The  value  of  r  is  given  in  Table  III.,  for  different  pressures. 

Let  us  assume  that  the  water  is  heated  under  these  condi- 
tions up  to  100°,  and  that  the  pressure  is  Ac  =  -p.  Now  let  heat 
be  still  further  imparted,  while  the  pressure  remains  unchanged. 
Vaporization  then  takes  place  under  constant  pressure  and 
temperature.  Suppose  we  thus  allow  a  certain  weight  of  water, 
X,  to  be  vaporized.  The  volume  is  increased,  and  Ac  is  carried 
to  BE.  Through  this  point  B  let  an  adiabatic  curve  be  con- 
structed.    The  heat  weight  necessary  for  vaporization,  which 

must  be  imparted  to  the  x  kilograms  of  water  is  -jjp'    Hence  the 

total  heat  weight  imparted  both  to  water  and  steam  is 

r         XT         c   ,  ,     T     ,    xr 

In  other  words,  this  equation  gives  the  heat  weight  neces- 
sary to  raise  1  kilogram  of  water  from  0"  and  the  corresponding 
pressure,  into  water  of  t  degrees  and  the  corresponding  press- 
ure, and  then  to  convert  x  kilograms  of  this  water  into  steam. 
As  soon  as  x  is  known,  we  can  find  the  corresponding  volume, 

V  =  XU  +   (J, 

and  ^an  then  lay  off  OE  and  EB. 

Suppose  again,  we  raise  the  water  from  0°  to  fi°,  for  which 
the  pressure  is  2\-  The  heat  weight  is  then,  under  the  assump- 
tion that  between  0°  and  ti°  the  mean  specific  heat  of  water  is 
the  same, 

Ti       c  ,  .273  +t,        c   ,  ,    T, 

^  =  ^lognat-^73-=^lognat2^. 

Now  let  a  certain  weight  Xi  of  this  water  be  vaporized  under 
constant  pressure  j^u  so  that  AF  passes  to  GH,  and  the  point 
G  is  in  the  adiabatic  curve.     The  heat  weight  imparted  to  the 

steam  is  -~  . 
ATi 

We  have,  therefore,  for  the  entire  heat  weight  imparted, 


ADIABATIO  CURVE  FOR  SATURATED  STEAM.  423 

Since  6^  is  a  point  in  tlie  adiabatic  curve,  tliis  lieat  weight 
must  be  equal  to  tlie  first. 
Hence 

A      AT      A'^  AT,' 


or 


r+^=r,  +  ^ (xxy.) 


The  values  of  r,  Tj,  and  -=,  ~,  are  given  in  Table  III.,  so 
that  Xi  can  be  easily  found  when  x  is  known.     "We  have 


xr\  Ti 


If  Xx  is  found,  the  volume  v^  is  given  by 

where  u,  is  given  by  pi  and  t^.  If  therefore  only  the  point  D  is 
given,  we  can  construct  the  point  G  on  the  adiabatic  through  D. 
In  similar  manner,  if  we  raise  the  kilogram  of  water  from  0° 
to  4°,  for  which  the  pressure  is  AI  =  p^,  the  heat  weight  added 
to  the  water  is 

-  =  ^lognat2-73. 

If  then  we  evaporate  x^  kilograms  under  constant  pressure, 

so  that  ^/passes  to  K,  the  heat  weight  is  -hfr  ^   ^^^l  we  have 

Al% 

for  the  total  heat  weight 

A       ATi' 


If  K  is  on  the  adiabatic. 


Tp-Ti-^      rp^     . 


^  +  77^  =  7-i  + 


424 


THERMOD  7NAMICS. 


SO  that  we  can  find  x-^  as  also  the  corresponding  volume  v^  =  oL. 
We  can  thus  construct  the  point  K.  Generally,  we  see  that  by 
the  principles  given  in  the  Appendix,  we  can  easily  find  dijBfer- 

ent  points  on  an  adiabatic.  For 
the  sake  of  illustration,  let  us 
take  a  special  example. 

Let  us  assume  that  we  have 
to  start  with  x=  0.80  kilograms 
of  steam,  and  hence  1  —  cc  =  0.20 
of  water.  The  pressure  ^  is  1 
atmosphere.  Then  we  have  for 
OE 

OE  =  v  =  xu+  ff. 

Since  for  j9  =  1,  w  =  1.65, 

V  =  0.80  X  1.65  4-  0.001, 

or,  disregarding  <J, 

V  =  0.8  X  1.65  =  1.320  cuk  m. 

'''•'•  ''•  •      Lay  off  then.  Fig.  72,  OE  = 

1.32  and  ED  —  1  atmos.,  and  D  is  a  point  in  the  adiabatic.   We 
may  construct  a  second  point  for  jj  =  '^ED  =  2  atmos. 
For  this  we  have 


r  + 


1+     rp^ 


In  Table  III.,  we  have  the  values  of  r,  r,  and  -™  for  different 


pressures. 
For^  =1 

hence 
and 


r  =  0.31, 


1.44, 


r  +  :^  =  0.31  +  0.8  x  1.44  =  1.46, 


T 


1.46  =  ri+xi^. 


ABIA BA  TIG  EXPANSI0N-8A  TUBA  TEB  STEAM.  425 


For  pi  =  2  atmos.     r^  =  0.37,  ^  =  1.33,  hence 


1.46  ^  0.37  +  X,  X  1.33, 
and 

1.46- 0.37  ^^_g,_ 


1.33 

By  tlie  rise  of  temperature,  0.02  kilograms  of  water  are  thus 
vaporized.  Now  Vi  =  cci^,  and  since  for  2^1  ~  2  atmospheres, 
«i  =  0.86,  we  have 

Vi  =  0.82  X  0.86  =  0.71  cubic  meters. 

Make,  then,  011=  0.71,  and  HG  =  2,  and  we  have  the  point 
G  of  the  curve. 
Let  j32  =  3  atmospheres.     Then 


r  4-  ^^  -  r   +  ^^^2 


1.46=r,  +  a^^. 


According  to  Table  III.,  for  ^2  =  3  atmospheres, 

r2  =  0.40,     ^  =  1.26 

hence 

1.46  =  0.40  +  X2X  1.26, 

and 

1.46  -  0.40      n  Q^  Ti 
X2  = z^-^. =  0.84  kilograms. 

Hence,  since  ii-^  =  0.59, 

V2  =  a'2^^2  =  0.84  x  0.59  =  0.50  cubic  meters. 


426  THERMOD  YNA  MIC 8. 

Lay  off  then,  OL  =  0.5,  and  LK  —  3,  and  ^  is  a  third  point  on 
the  curve. 

For  ^^3  =  4  atmospheres,  we  find  in  similar  manner,  %  =  0.38 
cubic  meters.  In  our  Fig.  72,  OB  =  0.38,  and  BC ~  4,  and  thus 
we  have  a  fourth  point  C.  The  curve  joining  these  points  is 
the  adiabatic. 

We  see  from  the  preceding,  that  when  a  mixture  of  steam 
and  water  is  compressed  adiabatically,  water  is  vaporized,  and 
at  the  end  there  is  more  steam  and  less  water  than  at  first.  If 
there  were  at  first  saturated  steam  only,  without  water,  by 
compression  the  steam  would  be  superheated,  and  the  adiabatic 
curve  for  this  superheated  steam  would  be  different.  If  the 
saturated  steam  expands  performing  work,  we  have  inversely, 
condensation  of  steam,  as  has  already  been  proved  elsewhere. 
The  deportment  of  saturated  steam  by  adiabatic  expansion  or 
compression  is  thus  the  reverse  of  that  for  the  isodynamic 
curve. 

Since  a  knowledge  of  the  law  of  the  adiabatic  curve  is  of  the 
greatest  importance  for  a  reliable  and  thoroughly  scientific 
theory  of  the  steam  engine,  we  shall  proceed  to  show  by  an  ex- 
ample, how  condensation  takes  place  during  expansion,  and 
shall  then  investigate  what  takes  place  when  we  have  at  first 
only  water  of  a  certain  temperature,  and  then  diminish  the 
pressure  according  to  the  adiabatic  curve. 

Suppose  in  a  cylinder,  1  kilogram  of  pure*  saturated  steam, 
without  admixture  of  water,  of  4  atmospheres  pressure,  and 
therefore  at  a  temperature  of  144"'.  Then  here  x  =  1,  and 
from  Table  III., 

r=  0.427,     1^=1.211, 

and  from  Table  11.,  u  =  0.447,  hence 

v^xu  =  l  X  0.447  =  0.447. 

If  now,  Fig.  73,  OB  =  0.447  and  ^(7=  4,  we  have  the  point 
C  as  the  first  point  of  the  curve.     "We  have  now 

r  +  a:  ^  =  0.427  +  1.211  -  1.638. 

*  "  Pure  "  i.  e  ,  dry— no  water  particles  being  meclianically  suspended  in  the  steam  When 
this  is  the  case  the  steam  is  said  to  be  "  wet.''' 


ADIABATIC  EXPANSION— SATURATED   STEAM. 


427 


Now  let   tlie    steam  expand  adiabatically,  until  the  pressure 
is  px  =  DE  —  2  atmospheres.     For  this  case 


Ti  =  0.368,    ^  =  1.326,  hence 

1.638  =  0.368  +  iCi  X  1.326,  or 
Xi  —  0.958  kilograms. 

Hence  1  -  0.958  =  0.042  kilo- 
grams of  steam  have  con- 
densed. The  volume  v^  —  x^u^ 
=  0.958  X  0.859  =  0.823.  If 
then,  OD  =  0.823  and  DE 
=  2,  E  is  a  second  point  in 
the  curve. 

Let  the  steam  still  expand, 
till  its  pressure  is  1  atmos- 
phere =  FG. 


Then 


r,=  0.314,^ 


1.438,     ^2  =  1.649,  and 


1.638  =  0.314  +  ^2  X  1.438     or    x^  =  0.920  kilogram. 

Hence  by  expansion  from  4  atmospheres  to  1, 1—  0.920  =  0.08 
kilogram  of  steam  have  been  condensed. 
The  specific  volume  is 


^2  —  X^U^ 


0.92  X  1.649  =  1.518  cubic  meters. 


If  we  make  OF  =  1.518  and  FG  =  1,  we  have  a  third  point 
in  the  curve. 

We  see  then,  very  plainly,  that  during  expansion  the  steam 
condenses.  If,  inversely,  we  had  to  start  with  only  0.920  (x^) 
kilogram  of  steam,  and  1  —  0.920  =  0.080  of  water,  under  a 
pressure  of  1  atmosphere,  and  compressed  the  mixture  adia- 
batically to  4  atmospheres,  we  would  have  at  the  end  of  the 
process,  1  kilogram  of  steam,  saturated,  and  the  0.08  kilogram 
of  water,  will  be  comj)letely  vaporized. 


428  THERMODYNAMICS, 

The  expansion  ratio  in  the  first  case  is 

V,      1.518 


0.447 


=  3.390. 


The  expansion  ratio  is  therefore  less  than  according  to  the 
old  views  as  to  the  properties  of  steam. 

Let  us  now  suppose  we  have  only  water  to  start  with,  of  a 
given  temperature,  the  pressure  being  therefore  such  that 
there  is  no  vaporization.  Now  let  the  pressure  diminish  gradu- 
ally, and  no  heat  be  imparted  or  abstracted,  and  let  us  see 
what  are  the  changes. 

Suppose  the  temperature  of  the  water  is  144^,  and  hence  the 
pressure  p  =  4i  atmospheres.     "We  have  then 


hence 


r:=  0.427,     1^=1.211,    and    x  =  0, 


T  +  x^  =  0A27. 


If  now  the  pressure  sinks  gradually  to  2  atmospheres,  we  have 

r,  =  0.368,     -^  =  1.326      and     t,  +  x,^=  0.368  +  x,  x  1.326, 

hence 

0.368  +  x^x  1.326  =  0.427,    and     x^  =  0.044  kilogram. 

This  weight  of  steam  has  been  formed.  If  in  this,  as  well  as  in 
the  previous  case,  we  had  used  more  decimal  places  and  cal- 
culated more  accurately,  we  would  have  found  that  the  same 
quantity  of  steam  was  formed,  as  in  the  case  of  pure  saturated 
steam  only  was  condensed. 
We  have  further 

Vi  =  x,u,  =  0.044  X  0.859  =  0.039  cubic  meter. 

If  the  pressure  still  falls  to  1  atmosphere,  we  have 

r2  =  0.314,     ^==1.438, 
0.427  =  0.314  4-  1.438x2,     or    x,  =  0.0786  kilogram. 


ADIABATIC  EXPANSION— SATURATED  STEAM.  429 

This  steam  has  been  formed,  and  there  is  left  1  —  0.0786 
=  0.9214  kilogram  of  water.  Here  also  we  should  have  just 
the  same  steam  weight  produced,  as  in  the  first  case  was  con- 
densed. 

We  see  then,  that  when  there  is  more  steam  than  water, 
there  is  partial  condensation  during  expansion.  But  when 
there  is  only  water  in  the  beginning,  steam  is  formed  during 
expansion.  Hence  it  follows,  that  there  is  a  certain  propor- 
tion of  steam  and  water  for  which,  during  expansion  there  is 
neither  condensation  nor  vaporization,  or  at  least,  for  which 
during  the  first  period  of  the  expansion,  there  is  just  as  much 
steam  generated  as  during  the  second  is  condensed.  This 
mixture  can  be  determined.  Since  at  the  beginning  and  end 
of  the  expansion,  we  have  the  same  amount  of  steam  or  water, 


(XXVI. 


If  we  assume  the  initial  pressure  at  4  atmospheres  and  the 
end  pressure  at  1  atmosphere,  we  have  for  x  almost  exactly  0.5 
kilogram.  We  must,  therefore,  have  to  start  with  as  much 
water  as  steam,  by  weight,  if  by  expansion  between  4  and  1 
atmospheres  there  is  to  be  at  the  end  the  same  steam  and 
water  quantity  as  at  the  beginning.  For  from  10  to  6,  and  5  to 
1,  and  1  to  I  atmospheres,  we  have  respectively 

X  =  0.56,         X  =  0.50,        X  =  0.46  kilogram. 

The  mixture  ratio  does  not  vary,  therefore,  much  from  1  to  1. 

If  we  suppose  for  the  extreme  pressures  4  and  1  atmospheres, 
and  the  mixture  ratio  1  to  1,  that  is,  as  much  steam  as  Avater, 
by  weight,  the  adiabatic  curve  and  also  that  for  constant  steam 
weight  constructed,  both  curves  must  then  cross  at  the  begin- 
ning and  end  of  expansion.  Fig.  74  Since,  further,  the  steam 
formed  during  the  first  half  of  expansion  is  small,  both  curves 
vary  but  little  from  each  other.     The  adiabatic  curve,  how- 


+  X 

T 

Ti  +  ^ 

X 

r 

-    ^1 

r 

n 

* 

T 

T, 

430 


THERMOD  YNAMIC8. 


ever,  approaches  tlie  axis  somewhat  more  rapidly  than  the 
curve  of  constant  steam  weight.  In  Fig.  74,  the  dotted  line 
is  the  adiabatic  curve  for  the  mixture  ratio 
of  1  to  1  and  for  the  limiting  pressures 
pi  4  and  1  atmospheres.  We  see  from  the 
preceding  how  complicated  is  the  phenome- 
non of  expansion  or  compression  adiabat- 
ically  of  saturated  steam.  We  thus  arrive 
at  the  followino;  general  results. 


(a.)  For  EXPANSION,  adiabatic : 

1.  1/  loe  start  tvith'  pure  saturated  steam, 
tuithout  admixture  of  ivater,  steam  condenses 
during  expansion!. 

2.  If  there  is  more  steam  than  loater,  there  is 
also  condensation. 

3.  If  there  is  more  ivater  than  steam  to  start 


kzsi     ! 


Fig.  74. 


with,  steam  is  generated  during  expansion. 

(b.)  For  COMPRESSION,  adiabatic : 

1.  If  ive  start  ivith  pure  saturated  steam,  without  admixture  of 
water,  it  ivill  be  superheated  by  compression. 

2.  //  the  initial  steam  iveight  is  greater  than  that  of  the  ivater, 
steam  is  generated  by  the  compression. 

3.  If  there  is  more  water  than  steam,  steam  is  condensed  during 
compression. 


Calculcdion  of  the  Outer  Work. — Since  during  the  expansion  or 
compression  according  to  the  adiabatic  curve,  heat  is  neither 
imparted  nor  abstracted,  the  outer  work  performed  during  ex- 
pansion must  be  at  the  expense  of  inner  work.  If  now  Z7is  the 
inner  work  contained  by  a  mixture  of  steam  and  water  before 
expansion,  and  Ui  that  after,  the  outer  work  is 

L=U-  Z7i. 

Hence  the  heat  disappearing  is 

Q  =  AL  =  A(U-  U,). 
The  inner  heat,  which  is  equivalent  to  the  inner  work,  is 


ABIABATIC  EXPANSION— SATURATED  STEAM.  43I 

easily  calculated.     If  we  liave  x  kilograms  of  steam  and  1  —  x 
of  water,  tlie 

X  kilograms  of  steam  contain ....  x{q  +  p)     lieat  units. 

1  —  X  "  "  water       "       (1  —  a:;)^'  " 

The  mixture  contains x{q  +  p)  +  {1  —  x)q  " 

or,  reducing, 

q  +  xp  lieat  units. 

For  X  kilograms  of  steam  and  1  —  x  of  water,  we  have 

Qi  +  ^iPi  heat  units. 

Hence  the  heat  disappearing  during  expansion  is 

Q  =  AL  =  A  {U-  Uy)  =  q  +  xp  -  {q,  -  x,p,).     .     (XXVII.) 

where  x  and  x-^  are  the  steam  weights  at  beginning  and  end  of 
expansion. 

Example  1.— What  work  is  performed  by  1  kilogram  of  saturated  steam  at  4 
atmospheres,  when  it  expands  adiabatically  to  1  atmosphere  ? 

"We  have  in  this  ease  x  =  1,  and  can  find,  as  on  page  425,  ic,  =  0.920.  Further, 
from  Table  11.,  we  have 

2  =  145.31,        and        p  =  461.5 

for  4  atmospheres,  and  for  1  atmosphere, 

g,  =100.5,        and        /3i=  496.3. 

Inserting  these  values,  we  have 

Q  =  AL  =  145.31  -  100.5  +  461.5  -  496.3  x  0.920 

=  49.7  heat  units,  and  hence 
X  =  424  X  49.7  =  21072.8  meter-kilograms. 

Example  2.— What  would  the  work  be,  if  to  start  with,  we  had  only  water 
and  no  steam  ? 

In  this  case  we  have  x  =  0,  and  find,  as  on  page  428,  x,  =  0.079,  while  q,  q,, 
p,  p,  are,  as  before.     Hence 

AL  =  145.31  -  100.5  +  0.461.5  -  496.3  x  0.079 
=  44.81  -  39.2  =  5.61  heat  units. 
L  =  434  X  5.61  =  2378.6  meter-kilograms. 


432  THERMODYNAMICS. 

The  work  is,  therefore,  very  mucli  less  than  before,  which  is 
easily  comprehended  from  the  fact  that  during  expansion  steam 
is  formed,  which  consumes  a  large  part  of  the  heat  of  the  liquid, 
so  that  the  temperature  and  pressure  decrease  more  rapidly 
than  before. 

Approximate  Formula  for  the  Adiabatic  Curve. — For  the  iso- 
thermal curve  for  steam  and  water  mixture,  we  can  easily  find 
the  volume  for  any  pressure  and  steam  weight.  For  the  iso- 
dynamic  curve,  also,  we  have  given  an  expression  which  gives 
with  sufficient  exactness  the  relation  between  pressure  and  vol- 
ume. Since  the  adiabatic  curve  is  of  especial  practical  impor- 
tance, it  is  desirable  to  find  for  it,  also,  such  an  expression, 
which  shall  furnish  us  with  a  simpler  and  less  tedious  method 
of  calculation. 

Eankine  found  that  the  law  of  the  adiabatic  curve  was  given 
by  an  equation  of  the  form 

where  m  =  1.11.  But  it  is  not  stated  by  Eankine  whether  this 
value  of  TO  was  found  for  every  mixture  of  steam  and  water,  or 
only  for  a  certain  definite  proportion.  From  the  calculations 
given  by  E-ankine,  the  first  appears  very  improbable,  and 
Grashof  has  pointed  out  that  this  value  is  too  small  for  pure 
saturated  steam  without  admixture  of  water.  He  shows  that 
in  this  case,  w  should  be  1.140.  The .  preceding  calculations 
enable  us  to  determine  at  once  if  this  value  is  correct,  and  at 
the  same  time  show  that  the  value  of  m  given  by  Eankine 
answers  only  to  a  certain  definite  mixture,  and  not  to  all  others. 
From  the  equation 

pv'^^^pivr (XXYIII.) 

we  have 


(XXIX.) 


Pi 

m  — 

'< 

lo.^^ 

ADIABATIG  CURVE— APPROXIMATE  FORMULA.         433 

Let  us  take  tlie  example  on  page  425. 

Here  ^  =  4,  pi  =  1,  v  —  0.447,  and  v^  —  1.520. 

Accordingly, 

_log4__  Jog4    __  0.602  _ 
1      1.520      log  3.89  ~  0.530  ~'^^' 
/°Sa447 

a  value  wHcli  agrees   perfectly  with  that  of  Grasliof  if  we 
take  only  2  decimal  places. 

If,  however,  we  suppose  only  water  at  the  beginning  of  ex- 
pansion, and  find  m  for  the  case  of  the  example  on  page  428. 

Here      p  =  4,  ^i  =  1,  v  —  0.001,  and  Vy  =  0.1312,  hence 
loff  4  log  4  0.602 


,      0.1312  "  log  131.2       2.118 

^°g-aoor 


=  0.284. 


■  It  follows,  then,  that  the  value  of  m  is  entirely  dependent 
upon  the  original  proportion  of  steam  and  water.  Zeuner  has, 
therefore,  calculated  m  for  different  mixtures,  as  follows  : 


Initial  Pressure 

P 
in  Atmospheres. 

Initial  Specific 
Steam  Quantity 

X. 

Final  Pressure  in  Atn 
0.5                       1 

4 

X  =  0.90 

m  =  1.124 

1.127 

0.80 

=  1.114 

1.116 

0.70 

=  1.103 

1.104 

2 

«=r0.90 

m  =  1.123 

1.126 

0.80 

=  1.114 

1.117 

0.70 

=  1.108 

1.104 

1 

X  =  0.90 

m  =  1.122 

0.80 

=  1.114 

0.70 

=  1.103 

2 

1.130 
1.119 
1.105 


We  see  from  this  tabulation  that  the  value  of  m  depends 
upon  the  original  steam  quantity  x ;  That 

1st,  it  is  greater  the  greater  x  is. 

2d,  it  depends  upon  the  initial  and  final  pressures.     The 
greater  these  are  the  greater  is  w. 
28 


434  THERMODYNAMICS. 

We  see  also  from  the  Table,  that  Rankine's  value  for  m  be- 
longs to  a  mixture  of  about  80  parts  steam  and  20  parts  water. 

The  dependence  of  m.  upon  the  initial  and  final  pressures  is 
also  shown  by  the  following  tabulation  given  by  Zeuner,  in 
which  the  steam  is  assumed  to  be  at  first  pure,  without  admix- 
ture of  water.     The  table  also  gives  the  expansion  ratio  e. 


Initial  Pressure 

p  Final  Pressure  y),,  in  Atmospheres, 

in  Atmospheres. 


0.5 

1 

2 

4 

X  - 

0.854 

0.884 

0.918 

0.956 

e  = 

11.577 

6.336 

3.375 

1.834 

m  = 

1.133 

1.136 

1.140 

1.143 

X  = 

0.888 

0.931 

0.958 

e  - 

6.283 

3.390 

1.837 

m  — 

1.133 

1.135 

1.140 

X  --= 

0.934 

0.960 

e  = 

3.409 

1.843 

tn  = 

1.130 

1.134 

X  = 

0.961 

e  — 

1.848 

«i  = 

1.129 

We  see  from  this  that  for  the  same  initial  pressure  p,  the 
value  of  m  is  greater,  the  greater  the  final  pressure  pi.  We  see 
also  that  the  deviations  are  slight,  and  that  hence  we  can  take 
for  m  the  mean  value  m  =  1.135. 

The  expression  which  gives  the  adiabatic  curve  of  saturated 
steam,  originally  luithout  admixture  of  water,  is  therefore 

pyivu  _  p^y\  134  __  Pjjv/^^^  etc. 
Hence 

Py~\v)  ' 

or  the  expansion  ratio  e  is 

c  =  h.^  (P\^^  (P_Y    .    .    (^^^.) 

V       \piJ  \piJ 


ADIABATIU  CURVE— APPROXIMATE  FORMULAE.         435 

Now  from  the  table  on  page  433,  we  see  that  for  the  same 
initial  pressure,  m  is  less  the  greater  the  water  weight  in  the 
mixture.  If  we  take  the  mean  of  those  cases  where  the  initial 
steam  quantity  is  0.90,  we  have  m  =  1.125. 

The  mean  for  x  =  0.80  is  m  =  1.115  and  for  x  ^  0.70,  1.103- 
Hence  we  have, 

for  a:;  =  1       m  =  1.135 

a:  =  0.90 m  =  1.125 

x  =  0.80 m  =  1.115 

a;  =  0.70 m  =  1.103. 

Zeuner  has  found  that  these  values  are  given  very  closely  by 
the  empirical  formula, 

772  =  1.035  +  0.100.'??    .     .     .      (XXXI.) 

Grashof  has  assumed  in  his  investigation  of  the  steam  en- 
gine, m  —  1.125,  a  value  which  corresponds  therefore  to  a 
mixture  containing  ten  per  cent,  of  water.  We  shall  refer  to 
this  when  speaking  of  the  steam  engine.  We  would  only 
remark  here,  that  the  steam  passing  from  the  boiler  to  the 
cylinder  has  always  a  certain  amount  of  water  suspended  in  it 
mechanically ;  that  the  amount  of  this  water  depends  upon 
the  velocity  and  the  fierceness  of  ebullition  ;  that  also  in  long 
passages,  some  steam  is  condensed  and  carried  into  the  cylin- 
der. In  locomotives  the  water  weight  is  not  unfrequently  25 
to  30  per  cent. 

Tlie.  indicator  diagram,  of  the  steam  engine  confirms  rather  than 
contradicts  the  correctness  of  the  mechanical  theory  of  heed.  If  we 
compare  the  indicator  diagram  of  the  steam  engine  with  the 
isothermal  curve  for  gases,  which  is  given  by  Mariotte's  law, 
we  find  that  this  curve  deviates  but  little  from  the  curve  of  the 
diagram.  It  has  thus  been  asserted  that  the  steam  in  the  cyl- 
inder of  a  steam  engine  follows,  during  expansion,  Mariotte's 
law,  pv  =  piVi,  and  that  hence  the  conclusions  of  the  mechanical 
theory  of  heat  must  be  incorrect.  Properly  regarded,  the  con- 
trary is  the  case.     The  value  of  m  in  the  equation 


436  THERMODYNAMICS. 

approaches  unity  more  nearly,  the  greater  the  quantity  of 
water  and  the  greater  the  expansion.  As  the  steam  always 
carries  with  it  a  considerable  percentage  of  water,  m  cannot 
differ  much  from  1,  and  hence  the  indicator  curve  does  not 
vary  much  from  that  which  gives  the  law  of  Mariotte.  If,  for 
example,  we  refer  to  the  figure  on  page  424,  we  find  that  between 
4  and  1  atmospheres  and  for  20  per  cent,  of  water,  the  volumes 
are  nearly  inversely  as  the  pressures.  Thus  while  these  last 
are  4,  2,  1,  the  former  are  0.38,  0.71,  1.32. 

Worlc  of  Steam  Ex^janding  AdiabaiicaUy. — We  have  seen  in 
Part  I.,  that  the  work  of  one  kilogram  of  air,  when  expanding 
adiabatically,  is 


^=.-^[^-(^)'] 


where  k  =  1.41,  and  the  equation  of  the  curve  is 

Since  our  equation  for  saturated  steam  has  the  same  form, 
we  have  a  similar  expression 


ni  —  1 

^  =  l^[l-(f)"']--(™^- 


If  we  compute  according  to  this  formula  the  work  during 
expansion  of  steam  for  different  initial  pressures  and  expansion 
ratios,  and  compare  the  results  with  those  given  by  the  for- 
mula on  page  431,  we  find  a  very  satisfactory  agreement.  We 
will  show  this  by  an  example,  which  will  serve  at  the  same 
time  to  illustrate  the  use  of  the  last  formula. 


EXAMPLE. 

What  is  the  work  done  by  1  kilogram  of  saturated  steam,  while  expanding 
adiabatically  from  4  atmospheres  to  1  atmosphere? 

Here  p  =  4,   i^i  —  1   and  v  =:  xu  +  (5  —  0M8,     Hence  —  —  0.25,   and  for 

,  we  have  ^\.-.^  =  0.122.     Substituting  these  values  we  have  * 

m  1.135 


ABIABATia  EXPANSION— WORK  OF  STEAM.  437 

4  V   0  447 
L  =  10334  X       Q^^gg     [1  -  (0.25)"-i22].     Now  log  (0.25)"  1=2  =  0.122  log  0.25 

==  7.9266,  and    (0.25)0  1=-'  ^   o.845.     Hence    1  -  (0.25)o i'^-'  =  0.155.     Further, 

4  V  n  447 
10384  X        ^    -^     =  136408.     Therefore 

L  =  136408  X  0.155  =  21143  meter-kilograms. 

The  corresponding  heat  is 

Q  =  AL=  ^1^  =  49.86  heat  units. 
424 

On  page  431,  we  found  49.7  heat  units,  a  very  close  correspondence. 

Zeuner  has  investigated,  by  several  other  examples,  how  far 
the  results  of  the  present  formula  agree  with  those  given  by 
the  previous.  The  following  tabulation  gives  the  comparison. 
The  product  AL^  indicates  the  result  of  our  present  formula, 
and  AL  that  of  the  other,  while  e  is  the  expansion  ratio. 


Initial  Pressure 
in  Atmospheres 

Final  Pressure  i^i  in 

.  Atmospheres. 

P- 

0.5 

1 

2 

4 

e  =  11.51 

6.247 

3.393 

1.843 

'8 

AL,  =  94.90 

74.02 

51.35 

26.73 

^Z  =  94.93 

73.75 

51.01 

27.57 

e=r    6.25 

3.392 

1.843 

4 

ALi  =  70.95 

49.22 

35.63 

AL  =  71.14 

49.17 

25.53 

e=    3.392 

1.842 

3 

AL,  =47.19 
AL  =  47.40 

e=    1.842 

24.57 
24.59 

1 

^i,  =23.58 
^i;  =  23.70 

We  see  that  the  values  of  AL^  and  AL  coincide  as  near  as 
can  be  desired.  "We  may  therefore  make  use  of  the  approxi- 
mate formula  XXXII.,  in  calculating  the  work  of  expansion  in 
the  steam  engine.  This  we  shall  do,  but  we  may  give  it  a  more 
convenient  form. 


438  THERMODYNAMICS. 


m  —  1 
m  /  ^,  \  w  —  1 


Thus  instead  of  ( -—  j         we  can  put  f  —  j         ,  and  there- 
fore 

As  already  remarked,  Grashof,  in  his  discussion  of  the  steam 
engine,  has  taken  m  =  1.125. 


APPENDIX  TO  CHAPTER  XVIII. 

The  principle  that  in  every  reversible  cycle  process,  the  pro- 
duct   of     the 

highest      and    ^^  ^  i 

lowest  abso- 
lute tempera- 
tures is  equal 
to  the  product 
of  the  inter- 
mediate abso- 
lute tempera- 
tures, the  cor- 
rectness of 
which  we  have 
shown  by  an 
example  on 
page  240,  of 
Part  I.,  can 
be  proved  gen- 
erally in  an  elementary  manner. 

Let  AA  and  AiA^,  Fig.  75,  be  adiabatic  curves;  TcT^  and 
T-aTs  lines  which  follow  the  general  law 

n  n 


PxVi' 


P%W 


Let  T  and  T^  on  A  A  be  the  absolute  temperatures  of  1 
kilogram  of  air,  and  T^  T-^  the  final  absolute  temperatures 
when  this  weight  of  air  passes  along  TcT^  or  T^aT^.  Let  the 
curves  TdT  Oswdi  T^gT^  be  isothermals. 

439 


440  THEBMODTNAMICS. 

Let  US  first  consider  the  curves  TcT.,  T^T^T,  and  TdT. 
For  the  isothermal  curve  TdT,  we  have 

pv  =  qw (1). 

For  TzT^T,  we  have 

Pi'^i'  —  <iw^ (2). 

For  TcTi,  we  have 

n  n 

pv'^  =PiV.i"' (3). 

Let  us  seek  first  to  determine  from  T,  T^,  and  n,  the  volume 
Oio  =  w. 
From  (2),  we  have 

Pi^i  X  ^/~^  =qio  X  io^~'^. 

Since,  however,  for  perfect  gases  p-iV-^  =  RT^,  and  qio  =  RT, 

RT^  X  v/-^=  RT  X  tu''-\ 
or 


-.(if)'-' (4). 


Let  us  now  find  v  in  terms  of  the  same  quantities. 
From  (3), 

— -1  — -1 

pv  ><■  V"^        =P2^i  X  V/'*       . 

Since  here  pv  —  RT,  andj:)^^^  =  ^T^, 


i^I^x  v'^"'=  RT,  xv.i^'\ 


v=vjm'-' (5), 


Hence 


T 

Now  we  have  seen  in  Part  I.,  page  187,  the  expressions 

AT,      °^     AT, 

are  very  appropriately  called  "  heat  weights."     But  Qi  was  the 


APPENDIX  TO  CHAPTER  XVIIL  441 

heat  imparted  upon  the  path  Ti  T^,  and  Qi  that  abstracted  upon 
the  isothermal  TiTi  (see  Figure  in  Chap.  VI.,  Part  L).  All  the 
heat  imparted  went  to  produce  outer  work.  Tx  was  the  abso- 
lute temperature  at  which  heat  was  imparted,  and  T^  that  at 
which  it  was  abstracted.  Let  us  apply  this  to  our  present 
Figure. 

Thus,  the  expression  -^^  is  the  heat  weight  which  must  be 

imparted  to  the  unit  of  weight  of  air,  when  it  passes  from  the 
condition^,  v,  T,  along  the  isothermal  TclT,  to  the  condition 
q,  w,  T.     Denote  this  heat  weight  by  P,  then 

P--§^ (6). 

Now,  according  to  Equation  XVI.  of  Part  I.,  page  158,  the 
heat  Q  imparted  is 

^  =  2.3026^P!riog-, 
or  using  natural  logarithms, 

(3=^PI^lognat-. 
Substitute  this  in  (6),  and  we  have 

P  =  J?  log  nat  - ,    (7). 

Here,  then,  is  a  new  expression  for  the  heat  weight.  It  is 
given  by  the  initial  and  final  volumes,  during  expansion  along 
the  isothermal,  and  by  P.  The  value  of  P  varies  for  different 
bodies. 

Now  we  have  also  found,  page  145  of  Part  I., 

c^^-1). 
^-      A       ' 


hence 


P=£(*Lpi),ognatL^    ....     (8). 


442  THERMODYNAMICS. 

If  we  put  now  for  lu  and  v  their  values  from  (4)  and  (5),  we 
have 

F  =  ^-(^  log  nat  -^^^=  ^  iJc-1)  log  .at(P)'--" 
<T 


P  = 


i  log  nat  (I') 


finally, 


1     mh  —  n    ,  >  (T-i 

c  log  nat 


A     m-n         ^         \T 


From  Equation   XLI.,    page   198,  Part  I.,  •  c  is  the 

specific  heat  s  for  the  law  of  change 

n  n 


Therefore  we  have 


P=-^lognat|^     ,,....     (9). 


This  is,  therefore,  the  heat  weight  for  the  path  TcT,. 

We  can  thus  find  the  heat  weight  for  any  transference  of  a 
body  from  one  condition  to  another,  if  we  know  the  initial  and 
final  temperatures  as  well  as  the  specific  heat  for  such  transfer. 
If  the  initial  temperature  is  t,  and  the  final  U,  then  T  =  273  +  t 
and  Ta  =  273  +  t^,  so  that 

„        s  ,  ,273  +  ^2 

^  =  Z1«S^^^  2-73^7- 

If  we  change  the  condition  T^,  p^,  v-i  along  the  same  curve 
J'acJ'into  the  condition  T,  2^,  v,  by  abstracting  heat,  the  heat 
weight  abstracted  is 


log  nat 


APPENDIX  TO  CHAPTER  XV III.  443 

If  ThNis  another  curve,  which  has  the  same  initial  point  T, 
and  whose  end  lies  in  the  adiabatic  A^A^,  then,  if  the  change  of 
condition  follows  this  curve,  for  which  the  specific  heat  is,  say 
.Si,  and  the  final  temperature  T^,  we  have 


log  nat 


T, 


Since  N  is  on  the  adiabatic  curve  A-^A^,  the  heat  weight 
added  along  TcT-^  is  the  same  as  that  along  TbN,  and 


log  nat  7p'  =  -X  log  iiat  -^ . 


Therefore,  when  a  hody  passes  from  the  same  initial  condition 
into  different  final  conditions,  the  heat  iceight  imparted  is  always  the 
same,  if  these  fined  states  are  in  the  same  adiabatic. 

On  page  175,  Part  I.,  we  have  seen  that  the  heat  Q  and  Q^, 
which  must  be  imparted  to  a  body,  when  passing  by  different 
isothermals  from  one  adiabatic  to  another,  are  as  the  abso- 
lute temperatures  T  and  T^.     Hence 

9.=  9l.       or      _?_  =  _?!_ 
T      T,  AT      AT,  ' 

That  is,  the  heat  weights  are  equal.    If,  now,  T^a  T3  is  a  curve 

whose  law  is  pv'"-  —  p-yV-f-  =  etc.,  that  is,  if  it  is  a  curve  of  the 
same  kind  with  TcT^,  we  have  again 

•j^     or     i?  log  nat  — =  ^  log  nat  ^. 

For  since  the  curve  T^aT^  is  of  the  same  kind  as  TcT^,  we 
have  the  same  specific  heat  s. 
Since  now 


^1     _    ^    _   ^  1... ...  ^2 


log  nat 


we  have 


AT,       AT     A^         T 


-^  log  nat  ^  =  ^  log  nat  ^   .    \     .     (10). 


444 


THERMOD  YITAMIC8, 


T       T^ 

From  this  equation  we  liave  -—=  j~-, 

or 

and  this  is  the  principle  ah^eady  deduced  in  Part  I.  for  a  spe- 
cial case. 
If  in  the  cycle  process,  TcT^,  T^T^,  T^aT^,  TiT,  we  consider 

the  heat  weight  imparted  on  the  way  TcT.^,  that  is,  -^  log  nat 

T' 

-j^ ,  as  positive,  and  that  abstracted  on  the  way  T^^aTx,  viz., 

s  T 

-J-  log  nat  ~ ,  as  negative,  we  can  say  that 


In  every  reversible  cycle  process  the  algebraic  sum  of  the  heat 
iveights  is  zero. 

We  have  thus  far  spoken  only  of  such  reversible  cycle  pro- 
cess as  are  inclosed  by  four  curves.  We  may  have  a  process 
composed  of  three,  or  more  than  four  lines.  One  of  more  than 
four  lines  is  called  compound.     In  Fig.  75,    TcT^T^TdT  is  a 

cycle  process  of  only 
three  lines.  In  such  a 
process,  also,  the  heat 
weight  imparted  is  equal 
to  that  abstracted.  For 
the  heat  weight  on  the 
way  TdT  is  equal  to  that 
on  the  way  TcT^. 

Since  we  shall  have 
frequent  occasion  to 
speak  of  "heat  weight," 
let  us  by  some  examples 
•X  endeavor  to  get  a  clear 
idea  of  it. 

Suppose  in  a  cylinder 
the  unit  of  weight  of  air  of  the  volume  i;,  and  absolute  temper- 
ature T,  and  pressure  p,  Fig.  76,  and  let  us  heat  this  air  under 
constant  volume  v,  until  the  pressure  is  p^  and  temperature  T^. 


A 

r« 

j 

\. 

.a 

1 

\ 

Pi 

"^A 

e--v-— \ 
6 

IT ^ 

APPENDIX  TO    CHAPTER   XVIII. 


445 


Tlie  lie  at  weight  imparted  is 


losr  nat 


T, 


where  c  is  the  specific  heat  for  constant  volume. 

Now  let  the  air  expand  under  constant  pressure  p^,  until  the 
volume  is  Vi,  and  temperature  T^.     The  heat  weight  added  is 

^  log  nat  ^^, 

where  ck  is  the  specific  heat  for  constant  pressure. 

The  sum  of  the  heat  weights,  during  the  change  of  condition 
from  p,  V,  T  to  v^,  pi,  T.^,  is  then 

■J  log  nat  ^  +  ^  log  nat  ^ , 

We  should  have  to  add  the  same  heat  weight  when  the  air 
is  brought  in  any  other  way  from  the  initial  condition  2^}  v,  T 
to  pi,  Vx,  T-i.  For  if  we 
suppose  an  adiabatic 
through  T^,  as  AA,  the 
heat  weight  from  T  to  c 
is  the  same  as  from  T  to 
Ti,  and  from  Ti  to  c  the 
same  as  from  T2  to  Ti. 
Hence  the  sum  along  Tc 
and  cT-i  is  equal  to  that 
along  TTi  and  T^T^. 

Again,  let  Os,  Fig.  77, 
be  the  specific  water  vol- 
ume (volume  of  1  kilo- 
gram of  water)  at  0°.  Let  sA  be  the  pressure  of  the  steam 
at  0".  Let  now  the  water  be  heated  until  the  temperature 
has  risen  to  f,  and  at  the  same  time  let  the  pressure  be  so 
increased  that  at  every  instant  it  is  equal  to  the  steam  press- 
ure which  looidd  exist  ivithout  this  outer  pressure.  Then,  if  the 
mean  specific  heat  of  the  water  between  0°  and  t°  is  denoted  by 
c,  we  have  for  the  heat  weight  imparted 


—X 


c  ,  ,f  +27B 

^  log  nat  Q— 2^3 


log  nat 


273* 


446  THERMODYNAMICS. 

If  now  we  suppose  tlie  pressure  constant  and  lieat  tlie  water, 
then,  as  we  know,  steam  is  formed  at  constant  temperature  t. 
If  after  a  certain  time  a  certain  volume  of  steam,  SD,  is  formed, 
the  line  BC  represents  the  law  of  change.  This  line  is  the 
isothermal  line  therefore  of  saturated  steam,  because  it  gives 
the  relation  between  volume  and  pressure  for  constant  tempera- 
ture.    If,  then,  the  amount  of  heat  imparted  from  B  io  0  is  Q, 

we  have  for  the  heat  weight  -rjj^  where  T  is  the  constant  abso- 
lute temperature.  This  heat  Q,  is  the  latent  heat.  When  the 
water  is  evaporated  at  100"  it  is  537  heat  units  per  kilogram. 
If,  then,  SD  —  BC  represents  the  volume  of  only  one  half  of  a 

537 

kilogram  of  steam,  Q  would  be  -^  ~  268.5  heat  units,  and  the 

heat  weight  imparted  is 

268.5   ^  424  x  268.5 
AT  100  +  273  • 


Hence  to  form  from  water  at  O'',  in  the  manner  described, 
a  certain  volume  of  saturated  steam  of  T,  requires  the  heat 
weight 

-^lognat2^+^^. 


If  we  suppose  the  vaporization  had  taken  place  according  to 
the  law  represented  by  AcC,  the  heat  weight  imparted  is  still 
the  same.  For  if  we  pass  through  B  an  adiabatic  A^A^,  we 
have  for  the  heat  weight  from  ^  to  c  the  same  value  as  from  A 
to  B,  and  from  do  C  the  same  value  as  from  B  to  C.  If  fur- 
ther, ACx  is  parallel  to  OX,  then  ACi  is  the  isothermal  for 
vaporization  at  O''.  The  heat  weight  imparted  from  A  to  (7i 
must  be  equal  to  that  from  A  to  B,  or  A  to  c. 

li  ACx  =  SDi  represents  a  volume  whose  weight  is  cc^  kilo- 
grams, and  if  the  latent  heat  of  vaporization  is  Ti,  then  the  heat 
weight  imparted  from  A  to  Ci  is 


A  X  273 


APPENDIX  TO  CHAPTER  XVIIL  447 

We  liave  tlien 

-^lognat  - 


J.     ^         273       ^  X  273 ' 


c  log  nat  — 


273       273  ° 

Since  c  is  known  and  r-i  can  be  found  for  every  temperature, 
we  can  find  from  this  equation  cca  for  every  value  of  T. 


QUESTIONS  rOH  EXAMINATION. 

What  fs  an  fsothcrmal  curve  ?  What  is  the  form  of  this  curve  for  saturated  steam  ?  Define 
what  is  meant  by  saturated  steam  ?  What  effect  has  the  addition  of  heat  to  a  mixture  of  water 
and  steam,  when  the  pressure  is  constant  ?  If  saturated  steam  is  compressed  while  the  tempera- 
ture is  Ivept  constant,  what  happens?  What  is  the  expression  for  the  outer  worlc  during  iso- 
thermal expansion  ?  What  for  the  lieat  added  ?  What  does  u  denote  in  our  notation  f  What  is 
f^pecific  volume  of  steam  ? 

What  is  an  isodjniamic  curve  ?  Make  out  its  equation  for  steam.  Show  how  to  construct  it. 
When  water  and  steam  expand  isodynamically  what  takes  place  ?  What  is  the  outer  work 
durinjr  expansion  ?    What  is  the  heat  imparted  ? 

What  is  an  adiabatic  curve  ?  ISIake  out  its  equation  for  steam.  Show  how  to  construct  it. 
In  adiabatic  expansion  of  saturated  steam,  if  there  is  no  water  at  the  beginning,  what  takes 
place  ?  If  there  is  more  steam  than  water,  what  tal^es  place  ?  Does  this  mean  more  by  volume  or 
by  weight  ?  If  there  is  more  water  than  steam,  what  takes  place  ?  In  adiabatic  compression,  if 
there  is  no  water  at  the  beginning,  what  takes  place  ?  If  the  initial  steam  weight  is  greater  than 
that  of  the  water,  what  takes  place?  If  there  is  more  water  than  steam,  what  takes  place? 
What  is  the  expression  for  the  outer  work  ?  What  for  tlie  heat  ?  What  is  the  form  of  the 
equation  for  the  adiabatic  curve  for  saturated  steam  ?  What  is  the  value  of  the  exponent  for 
dry  steam  alone  ?  What  for  0.90  per  cent,  steam  ?  For  0.80  per  cent.  ?  0.70  per  cent.  ?  What 
is  the  general  equation  which  gives  the  value  of  the  exponent  for  any  percentage  of  steam  ? 
What  does  the  indicator  diagram  seem  to  show  as  regards  the  adiabatic  curve  of  saturated 
steam  ?  What  is  the  expression  for  the  work  of  saturated  steam  expanding  adiabatically  ? 
What  for  the  heat  ?  What  two  formuloe  have  we  then  for  this  work  ?  Do  these  agree  in  their 
results  ? 

Can  j'ou  prove  generally  that  in  every  reversible  cycle  process  the  product  of  the  highest 
and  lowest  absolute  temperatures  is  equal  to  the  product  of  the  intermediate  absolute  tempera- 
tures ?  What  do  you  understand  by  heat  weight  ?  What  is  the  specific  heat  for  any  law  of 
relation  of  pressure  and  volume  f  What  is  the  general  expression  for  the  heat  weight?  If  a 
body  passes  from  the  same  initial  condition  into  different  final  conditions  upon  the  same  adia- 
batic, what  IS  the  relation  between  the  heat  weights  imparted  ?  In  every  reversible  C5'cle  process, 
what  is  the  algebraic  stun  of  the  heat  weights  ?  What  is  a  cycle  process  ?  When  is  it  reversible? 
When  simple  1    When  compound  ?    When  complete  ?    When  incomplete  ? 


CHAPTEE  XIX. 

OTHER  CHA2JGES    OF    CONDITION    OF    STEAM    AND    LIQUID   MISTUEES, 
OF  PEACTICAL  IMPORTANCE. 

I.  The  Deportment  of  Steam  and  Liquid  Mixtures,  luhen  Heat  is 
Imparted  or  Abstracted  under  Constant  Volume. — Suppose  1  kilo- 
gram of  steam  and  liquid  inclosed  in  a  vessel,  of  whicli  x  kilo- 
grams are  steam  and  1  —  x  liquid.  The  temperature  of  tlie 
mixture  is  t  and  tlie  pressure  p.  The  specific  volume  v  of  the 
mixture  is  then 

V  =  XU  -{-  ff. 

Let  now  heat  be  imparted  or  abstracted.  The  temperature 
is  then  ^i,  the  pressure  2h,  and  the  specific  steam  quantity  is  x^. 
For  the  final  specific  volume  we  have 

V^  =:  X{Ui  +   6. 

If  now,  the  volume  is  kept  constant  oxv~  Vu 
xit  +  a  =  XiUx  +  c, 
hence  the  weight  of  steam  at  the  end  of  the  operation  is 

x^  =  -x  .    .    '.    .    .    (XXXIY.) 

EXAMPLE. 

Suppose  We  have  in  a  Vessel  a  mixture  of  G  kilograms  of  water  and  steam  of. 
which.  0.8(9  are  steam,  and  0.26^  water.  The  pressure  is  1.5  atmospheres.  "What 
will  be  the  specific  steam  volume  x-^^  when  the  steam  has  lost  so  much  heat  that 
the  pressure  is  only  -iVth  of  an  atmosphere  ? 

448 


STEAM  AND   WATER  MIXTURES— CONSTANT  VOLUME.    MQ 

For  U  atmospheres  v,=    1.126  (Table  II.) 

"    -,-V  "  ,       M  =  14.551, 

hence 

X,  =  —^  X  0.8  =  0.0774  X  0.8  =  0.06193  kUograms. 

Hence  the  entire  steam  quantity  at  the  end  is  0.0619  6^  kilograms,  and  the 
water  quantity  is  (1-0.0619)6^  =  0.93816^  kilograms.  There  is  then  0.8- 
0.0619  =  0.73816^  kilograms  of  steam  condensed. 

What  is  the  amount  of  heat  Q  which  must  be  imparted  or 
abstracted  from  the  mixture  per  kilogram  ? 

Since,  in  this  case,  there  is  no  outer  work,  all  changes  affect 
only  the  inner  work.  If  then  q  is  the  heat  of  the  liquid,  p  the 
inner  vaporization  heat  at  the  beginning,  and  q^  and  pi  at  the 
end, 

Q  =  q-  qi  +  p^-  Pi^i 

or 

^  '  u 

Q  =--q-qx  +  px-  p^-x 


=  q-qx 


+-(i-S-  •  (^^^^•) 


EXAMPLE. 

What  is  the  amount  of  heat  abstracted  in  the  preceding  example  ? 
For  li  atmospheres 


For  -/u  atmosphere 
Hence 


q  =  112.41  ^  =  438. 


91  =  46.28  -^  =  37, 


Q  =  112.41  -  46.28  +  0.8  x  1.126  (433  -  37)  =  66.18  +  0.9  x  896 
=  422  heat  units  per  kilogram. 

The  preceding  finds  application  in  the  condenser  of  a  steam 
engine.  Let  AB,  Fig.  78,  be  the  steam  cylinder,  C  the  piston,  EF 
the  condenser  communicating  with  J.  J5  by  the  pijoe  and  cock  JD. 
In  steam  engines,  usually,  communication  of  the  cylinder  with 
the  condenser  is  opened  before  the  end  of  the  stroke.  Thus, 
for  example,  when  the  piston  is  at  C,  and  still  going  up,  com- 
29 


450 


THERMOD  YNAMIG8. 


munication  may  be  opened.  Immediately  tlie  liot  steam  of 
]iigli  pressure  in  the  cylinder  mixes  with  the  colder  steam  of 
lower  pressure  in  the  condenser,  so  that  al- 
most at  once  there  is  a  medium  pressure  in 
both  vessels.  We  shall  see  presently  how  to 
calculate  this  medium  pressure.  Since  now 
the  condenser  is  kept  always  cool  either  by 
water  applied  on  the  outside  (surface  conden- 
sation), or  by  water  injected  (jet  condensation), 
this  mean  pressure  quickly  sinks,  owing  to 
the  abstraction  of  heat,  and  we  may  assume 
that  this  fall  of  pressure  takes  place  while  the 
piston  hovers  at  the  end  of  its  stroke.  Heat  is 
thus  abstracted  under  constant  volume.  The 
steam  in  the  entire  space  (cylinder  and  con- 
denser) has  now  only  the  pressure  correspond- 
ing to  the  temperature  of  the  condenser,  and 
the  cylinder  sides  are  kept  hot.  This  was,  as  we  know,  the 
reason  which  led  Watt  to  employ  separate  condensation,  and 
it  constitutes,  indeed,  the  chief  value  of  his  discovery,  regarded 
from  an  economical  stand-point. 

The  question  arises,  how  many  kilograms  of  cooling  water 
are  necessary  for  each  kilogram  of  mixture  in  our  last  example, 
if  this  water  is  heated,  from  say,  t  =  15°  to  t^  =  35°  ?  We  have 
from  Equation  II.,  page  376,  for  the  heat  of  the  liquid  at  15°, 
q'  =  15.005,  and  for  that  at  35°,  q"  =  35.037.  If  Gi  kilograms  of 
water  are  required  for  abstracting  the  422  G  heat  units,  we  have 


G,  (35.037  -  15.005)  =  422  G^, 


and  hence  the  ratio  of  the  weight  of  cooling  water  to  that  of 
the  steam  is 


422 


422 


35.037  -  15.005  ~  20.032 


21. 


Let  us  now  investigate  what  amount  of  heat  should  be 
imparted  to  a  mixture  of  steam  and  liquid  in  order  that  the 
pressure  and  temperature  may  rise  to  a  certain  point. 

Let   the   initial  steam  weight  be  asrain  xG  kilograms,  the 


STEAM  AND  WATER  MIXTURES-CONSTANT   VOLUME.     451 

water  weiglit  (1  —  x)G  kilograms.  At  tlie  end  of  tlie  lieat  ad- 
dition let  there  be  Xi  G  kilograms  of  steam,  and  (1  —  x^  G  of 
liquid.  In  tlie  beginning  let  tlie  beat  of  tlie  liquid  be  q  and  tlie 
inner  latent  beat  p  ;  at  the  end,  gi  and  Pi.  Tben  we  bave  for 
tbe  amount  of  beat  imparted,  for  eacb  kilogram 

Q  =  q,  —  q  +  ux  {  —  ~  —  ]  beat  units, 
or  for  G  kilo.a,Tams 

beat  units. 


<?=[.-. .»(t-f)]^- 


By  tbe  aid  of  tliis  formula  we  can  solve  a  question  of  great 
practical  interest,  viz. : 

In  ivhat  time  ivill  the  j^ressure  of  steam  in  a  boiler  rise  hy  a  cer- 
tain amount  {say,  to  double  or  treble),  lohenfrom  a  given  instant  no 
more  steam  is  draiun  off  ? 

Let  us  assume  tbat  tbere  are  Q^  beat  units  imparted  every 
minute  to  tbe  boiler,  and  tbat  at  tbe  moment  of  closing  tbe 
valve,  tbere  are  xG  kilograms  of  steam,  and  after  Z  minutes 
tbere  are  x^^  G  kilograms ;  tbe  initial  pressure  being  jh  and  that 
after  Z  minutes  ^h-  Since  in  Z  minutes  ZQ^  beat  units  are  im- 
parted, we  bave 

ZQ^^Q^,     or    Z=^. 
If  in  23lace  of  Qx  we  insert  tbe  value  above,  we  bave 

This  expression  may  be  simplified.     From  page  449, 

C  1 

Z  =  -^  (gi  -  q  +PxXx  -  px)  =  -jr-  [(^1  —  q)G  +  {p^X^  -  px)  G'] 

(XXXYI.) 

Now,  ((/i  -  q)  G  is  tbe  excess  of  tbe  sensible  beat  over  tbat 
at  tbe  beginning,  and  {PiX^  —  px)  G  is  the  excess  of  the  inner 


452  •  THERMODYNAMICS. 

latent  lieat.  Since,  however,  Pi  and  p  differ  by  less  tlian  q^^  and 
q,  and  the  steam  weights  x  and  x-^  are  very  small  compared  to 
the  entire  weight  G  of  water  and  steam,  especially  when  we 
remember  that  the  boiler  is  about  |ds  full  of  water,  we  can 
neglect  {PiXi  —  px)  G  in  com23arison  with  {q^  —  q)  G,  and  then 

Z=-^{q,-q)    .     .     .      (XXXVII.) 

If  c  is  the  mean  specific  heat  of  water,  and  f^  and  t  the  initial 
and  final  temperatures,  we  have  very  nearly 

Z=-^{t,-t)c.     .     .    (XXXYIII.) 


This  very  simple  expression  shows,  that  the  time  in  which 
the  pressure  or  temperature  of  the  steam  in  the  boiler  rises  by 
a  certain  amount  is  proportional  to  the  weight  of  the  water,  or 
capacity  of  the  boiler,  as  well  as  to  the  difference  of  tempera- 
ture, and  inversely  proportional  to  the  heat  (Q^)  imparted. 

This  is  in  fact  evident,  for  it  is  clear  that  the  time  within 
which  the  temperature  rises  a  certain  amount  must  be  greater 
the  greater  the  weight  of  the  entire  mixture,  and  that  for  this 
rise  of  temperature  less  time  will  be  required  the  more  heat  is 
imparted  in  a  unit  of  time. 

The  heat  imparted  to  a  given  mass  of  water  in  a  unit  of  time 
dejDends,  however,  also  upon  the  extent  of  heating  surface, 
as  well  as  upon  the  intensity  of  the  heating.  The  first  consists, 
in  every  boiler,  of  two  parts,  the  direct  and  indirect  heating 
surface.  The  direct  heating  surface  is  that  directly  in  contact 
with  the  fire,  the  indirect  is  that  in  contact  with  the  heated 
gases.  Boilers  which  are  required  to  generate  steam  quickly, 
and  yet  not  hold  much  water,  such  as  locomotive  boilers,  etc., 
possess  a  relatively  great  direct  heating  surface.  The  same 
weight  of  water  thus  receives  in  the  same  time  a  much  greater 
amount  of  heat  than  in  boilers  where  the  direct  heating  surface 
is  small  in  comparison  to  the  indirect,  and  which  are  therefore 
larger  and  contain  more  water,  as  in  stationary  engines.  In  a 
locomotive  boiler,  therefore,  the  steam  pressure  rises  much 
quicker  than  in  that  of  a  stationary  engine. 

We  may  illustrate  the  preceding  by  an  example. 


STEAM  AND ,  WATER  MIXTURES— CONSTANT  VOLUME.    453 


The  heating  surface  of  a  cylindrical  steam  boiler  is  18  square  meters  (accord- 
ing to  ZeLiner,  it  is  about  15  H.  P.).  The  contents  are  11  cubic  meters,  of  which 
0.6  are  water  and  the  rest  steam.  When  the  engine  is  in  ordinary  action,  the 
boiler  generates  every  hour  25  kilograms  of  steam  for  every  square  meter  of  heat- 
ing surface,  of  5  atmospheres  pressure.  In  how  many  minutes  will  the  pressure 
rise  to  10  atmospheres  ? 
First  we  have  for  the  steam  weight  generated  per  minute, 

18  X  25       ^  ^  ,  ., 
— ^r^ =  7.5  kilograms. 


If  this  steam  is  generated  from  water  at  15%  we  can  easily  calculate  the  heat 
units  necessary  for  its  generation. 

Since  in  the  present  case  (when  the  machine  is  in  motion)  the  heat  has  to 
perform  outer  work  as  well  as  inner,  we  have  for  the  total  heat  which  must  be 
imparted  in  order  to  raise  1  kilogram  of  water  at  t  degrees  into  1  kilogram  of 
steam  at  t-^  degrees, 

W  =^  q^  —  q  +  r  =  q-i  —  q  +  p  -{-  Apu. 

In  our  case  t,  =  152.2,  q,  =  153.74,  q  =  15.005,  p  =  454.99,  and  Apu  =  44.19. 
Hence 

W=  153.74  -15.005  +  454.99  +  44.19  =  637.92  heat  units. 

For  7.5  kilograms,  therefore,  we  must  have 

7.5  X  637.92  =  4784.5  heat  units. 

This  is  then  the  heat  imparted  per  minute  to  the  water.  It  represents  Q,,,  in 
Equation  XXXVI.  Since,  further,  the  contents  of  the  water  space  is  0.6  x  11 
=  6.6,  the  water  weight  is  =  6600  kilograms.  The  steam  weight  can  be  disre- 
garded, and  we  have  thus  Cr  =  6600  kilograms.  If  now  we  take  c  =  1.022,  we 
have 

^=  ^^^(180.31  -  152.22)  X  1.022, 

where  180.31  is  the  temperature  of  the  water  at  10,  and  152.22  that  at  5  atmos- 
pheres.    We  have,  therefore, 

Z  =  39. 71  minutes. 

If  we  assume  that  a  locomotive  boiler  furnishes  in  the  same  time  3.3  times  as 
much  steam,  which  is  in  general  not  far  from  correct,  the  time  in  raising  the 
pressure  from  5  to  10  atmospheres  is  only 


454  THEBMODYNAMIGS. 

Fairbairn  found  the  time  required  to  raise  tlie  water  of  a 
locomotive  boiler  from  t  to  ii,  to  be  given  by  the  empirical  for- 
mula 

Z  =0.4:05  (t,-t), 

where  Z  is  the  number  of  minutes.     If  we  take  ti  ==  180.31  and 
t  =  152.22,  we  have 

Z  =  0.405  X  28.09  =  11.38  minutes, 

which  agrees  well  with  our  result. 

We  should  bear  in  mind,  that  the  steam  pressure  increases 
in  a  much  more  rapid  ratio  than  the  temperature,  and  hence 
in  a  boiler  with  high  pressure  less  time  is  necessary  for  a  cer- 
tain increase  than  in  a  boiler  with  low  pressure.  Thus,  for  in- 
stance, let  us  suppose  the  normal  pressure  in  the  preceding- 
example  to  be  9  instead  of  5  atmospheres,  and  then  see  how 
many  minutes  (Z)  are  necessary  for  a  rise  of  5  atmospheres, 
from  9  to  14.    We  have  now  for  f^,  195.53,  and  t  =  178.02 ;  hence 

6600    ^j^gg  g3  _  178.02)  x  1.022  =  24.7  minutes. 


4784.5 


or  much  less  time  than  in  the  first  case.  In  order  to  conclude, 
therefore,  whether  the  empirical  formula  of  Fairbairn,  applied 
to  locomotives,  gives  results  in  accord  with  our  formula,  we 
must  know  for  what  mean  pressures  it  is  made  out. 

It  is  also  easy  to  see  from  our  tables  that  an  increase  of 
pressure  of  double,  treble,  etc.,  takes  place  in  a  shorter  time 
for  low  pressures  than  for  high. 

The  preceding  calculations  show  that  the  time  in  which  the 
pressure  in  the  boiler  of  a  stationary  engine  increases  to  double 
or  treble  when  the  steam  pipe  is  closed,  and  the  steam  genera- 
tion is  as  when  in  use,  is  tolerably  great,  even  when  heat  is  im- 
parted as  during  ordinary  action  of  the  engine.  If,  when  the 
engine  is  not  in  action,  the  fire  is  left  to  itself  and  only  fed 
enough  to  keep  it  going,  the  time  will  be  much  greater.  In 
stationary  engines,  then,  there  is  less  danger  of  explosion  from 
this   cause.     In  locomotives  and  such  boilers,  which  have  a 


STEAM  MIXTURES. 


455 


large  and  direct  lieating  surface,  and  tlie  fire  is  kept  up  full, 
tlie  danger  is  greater. 

Mixture  of  Steam  Quantities  ivhen  in  Different  Conditions. — In 
tlie  preceding  we  liave   considered  the  cliange 
of  condition  of  a  mixture  of  steam  and  liquid, 
under  constant  volume.     Let  us  now  consider 


a.     -^^iJr-" 

of        m^ 

3c-   ^..:.-;fe^^?^^?g 


kind,  but  having  in  each  a  different  temperature, 
pressure,  etc. 

Let  A  have  the  volume  V,  and  B  the  volume 
Vi-     Both  vessels  are  connected  by  the  cock 
In  the  first  we  have  xG  kilograms  of  steam 
the  temperature  t  and  pressure  p  ;   in  the  sec- 
ond, a?i  Gi  kilograms  of  steam  of  the  temperature 
^1  and  pressure  j^jj.     The   water   in  A   is   then 
(1  —  x)  G,  and  in  B  {1  —  x-j)  Gi  kilograms.     As 
soon  as  the  cock  a  is  opened  the  steam  in  the 
vessels   mixes,  and   it  is  required  to  find  the 
final  condition  after  mixture,  when  heat  is  neither  added  nor 
taken  away. 

"We  have 

V  =  {xu  +  (j)G     and     Vi  =  (a?iMi  +  ff)  Gi. 
After  opening  the  cock,  the  total  volume  is 

V,=  V+  Ti  =  {xoji.  +  (t){G+  G,), 

where  Xo,  is  the  specific  steam  quantity  after  mixture. 
Hence 

(CCM  +  ff)  G  +  (a7i?«i  +  c)  6^1  =  {X.,212  +  0-)  ((r  +   Gi), 

or,  after  reducing 

{G+  Gi)  x,n,  =  Gxu  +  GiX^ii^ .    .     (XXXIX.) 

In   general,  Gi  may  be   expressed   in   terms   of   G,  so   that 
Gi  =  nG.     We  have  then 


(1  +  n)  x^u^  =  xu  +  nxiUx 


(XL.) 


456  THERMODYNAMICS. 

"We  liave  in  this  equation  two  unknown  quantities,  x.j  and  u^. 
We  must  establish  another  equation  between  them. 

Let  the  heat  of  the  liquid  in  A  be  q,  and  in  B,  q^,  and  the 
inner  latent  heats  p  and  pj.  Then  the  amount  of  heat,  measured 
in  heat  units,  in  the  first  vessel  is 

{q  +  xp)  G, 
and  in  the  second 

After  mingling,  let  the  heat  of  the  liquid  be  q^,  and  the  inner 
latent  heat  be  p2.     Then  1  kilogram  of  the  mixture  contains 

9.2  +  X2P2  heat  units. 

Since,  now,  we  had  in  A,  G,  and  in  B,  Gx  kilograms  of  water 
and  steam,  we  have  still  the  same  total  amount,  and  hence  the 
heat  contained  by  the  total  mixture  is 

{G  -V  G])  {q-i  +  x^p^  heat  units. 

Since  heat  is  neither  added  nor  abstracted,  we  must  have 

{G  +  Gx)  {q^  +  x^p^)  =  {q  +xp)  G  +  {qx  +  x^Pi)  G^. 
or,  putting  Gi=nG, 

(1  +  n)  (^2  +  Xip_)  =q  +  xp  +  {qx  +  x^Pi)  n. 


Hence 


g  +  a?P  +  (gi  +  XxP])n 


q,+^P^  ^_^,^ 


Since  all  the  quantities  on  the  right  are  known,  we  can  put 
the  expression  on  the  right,  for  the  sake  of  brevity,  equal  to  p, 
and  thus  have 

q-i  +  0C2P2  =  2^- 

Further,  from  Equation  XL., 

xu  +  nxMt 
^  ^  1  +  n 


STEAM  MIXTURES.  4:57 

or,  denoting  the  expression  on  tlie  right  bj  p]_ 


hence 


^^=^1', (XLI.) 

"2 


q2  +  -^ih  =  p.    .    .    .    (XLii.) 


From  Table  II.  we  can  find,  by  trial,  the  values  of  q^^  and 

—   in   order   that  q..  +  —  «.  =  p.      As   soon  as  qo  and  —  are 

known  we  can  find  at  once  the  pressure  2h  and  temperature  t^  of 
the  specific  steam  weight  ah,  that  is,  we  can  find  the  condition 
of  the  mixture  after  the  cock  is  opened.  An  example  will 
make  the  use  of  the  formula  clear. 


EXAMPLE. 

In  the  cylinder  A  we  have  G  kilograms  of  ptire  saturated  steam  at  1.5  atmos- 
pheres, and  in  J3,  G^  =  24.3806r  kilograms  of  water  and  steam  of  -,-'-11  atmosphere, 
of  which  0.0095  kilograms  are  steam.  What  is  the  condition  of  the  mixture  after 
the  cock  is  opened  ? 

For  the  pressure  1.5  atmospheres,  we  have  from  Table  II.,  <?=  112.41, 
p  =  487.01,  and  tt=  1.126. 

For  the  pressure  0.1  atmosphere  Ave  have 

^1  =  46.28,     p,  =  538.85,  and  m,  =  14.55. 

Also  w  =  24.38,  and  .Ti  =0.0095.     Hence 

1.126  +  24.38  X  0.0095  x  14.55      1.126  +  3.376      ^  ,„„ 
^3%  =P.  = 5^-^4:38 =         25.38         =  0-177. 


Also 


qz  +  x.p^ 


112.41  +  487.01  +  (41.28  +  0.0095  x  538.85)  x  24.38 


1  +  24.; 


=  72.99  =_p. 
Accordingly, 


^2  +  -^  x  0.177  =  72.99. 

Wo 

ISTow  from  Table  II.,  after  a  few  trials,  we  hit  upon  q^  —  60.59,  for  which 

—  =  70,  and  these  substituted  give 
Ms 

60.59  +  70  x  0.177  =  60.59  +  12.39  =  72.98, 


458  THEBMODTNAMIGS. 

a  value  almost  exactly  equal  to  72.99.  The  heat  of  the  liquid  {q.^)  after  the 
mingling,  is  then  60.59  heat  units.  Hence,  from  the  table,  the  temperature  ^, 
is  60.4°,  and  the  pressure  p^  is  0.2  atmosphere.  Since  for  this  pressure  ■Wg =7.542, 
we  have  from  XLI.,  for  the  steam  weight  in  each  kilogram 

X2—  —  =  ^t^f?.  =  0.0234  kilograms. 
Mg       7.542  "^ 

Before  the  mingling,  we  had  in  A,  Q  kilograms  of  steam  and  no  water,  and 
in_B,  a;,  G^i  =0.0095  X  24.386^  =  0.23166^  kilograms  steam  and  (1-0.0095) 
24.38(?  =  24.148(?  kilograms  of  water. 

In  both  vessels,  then,  the  total  steam  quantity  was 

(1  +  0.2316)(y  =  1.23166'  kilograms. 
After  mingling,  the  steam  weight  is 

xM-\-XiG^-  x.^Q{:X  +  24.38)  =  (?  x  25.38  x  0.0234  =  0.5966^  kilograms. 

Hence 

(1.2316  -  0.596)(?  =  0.63566^  kilograms 

of  steam  have  been  condensed. 

We  can  also  find  the  ratio  of  the  volumes  of  the  two  vessels.  If  the  volume 
of  J.  is  F  and  of  5  is  Fi ,  we  have 

V  \   Y^  =  G{xu  +6):  a,  {x,u,  +<?)...     (XLIII.) 
=  xu  +  6  :  24.38  {x^u^  +  6) 
=  1.127  :  3.388 
=  1  to  3  very  nearly. 

The  preceding,  together  with  what  has  been  said  on  page  448 
and  the  following,  forms  the  basis  of  the  theory  of  the  con- 
denser. We  have  already  referred  to  the  action  of  this  ap^^a- 
ratus.  We  noticed  that  the  cylinder  was  put  in  communication 
with  the  condenser,  when  the  piston  is  near  the  end  of  its 
stroke.  We  have,  then,  a  sudden  mixing  of  the  hot  and  high 
pressure  cylinder  steam  with  the  colder  and  lower  pressure 
condenser  steam.  The  above  calculation  includes  this  case. 
It  shows  how,  from  the  pressure  of  the  cylinder  steam  and  the 
condenser  steam,  we  can  find  the  mean  pressure  of  the  steam 
in  both  vessels.  This  pressure  then  falls  rapidly  down  to  that 
of  the  condenser,  by  reason  of  the  cooling  produced  by  the  jet 
or  by  the  surrounding  water.  The  calculations  in  the  first 
part  of  this  chapter,  then,  enable  us  to  calculate  the  heat  thus 
abstracted,  under  constant  volume. 


STEAM  MIXTURES.  459 

If  tlie  condenser  is  fitted  with  a  gauge,  we  sliould  see,  at  tlie 
moment  of  communication  witli  the  cylinder,  a  sudden  rise  of 
pressure,  lasting  but  for  an  instant,  and  then  a  quick  return  to 
the  condenser  pressure. 

It  will  now  be  easy  to  give  a  complete  theory  of  the  con- 
denser. There  are  two  kinds  in  use.  The  first  presents  a 
very  great  surface  which  is  continually  in  contact  with  water, 
so  that  it  is  kept  cool.  The  steam  is  condensed  without  coming 
into  direct  contact  with  the  water.  This  is  called  the  surface 
condenser.  The  other  consists  of  a  vessel,  not  only  sur- 
rounded by  cold  water,  but  into  which  cold  water  is  injected. 
The  steam  is  thus  condensed  by  direct  contact  and  mixture 
with  the  water.  It  is  called  the  jet  condenser.  In  the  first,  we 
have  only  to  cause  a  circulation  of  water  by  means  of  a  pump, 
and  by  means  of  another  pump  to  restore  the  condensed  steam 
to  the  boiler,  and  if  there  is  no  loss  by  leakage  of  steam,  we 
have  a  complete  cycle  process.  In  the  jet  condenser,  we  have 
not  only  to  remove  the  condensed  steam  but  also  the  water  in- 
jected. Since  this  is  in  weight  sometimes  more  than  20  times 
that  of  the  steam,  the  jjump  must  be  much  greater  than  for 
the  surface  condenser.  The  injection  water  also  contains  air, 
which  must  also  be  removed  by  the  pump.  For  this  reason  it 
was  called  by  Watt  the  "  air  pump." 


QUESTIONS  FOE  EXAMINATION. 

If  heal  is  imparted  to  or  abstracted  from  a  mixture  of  steam  and  water,  the  volume  of  which 
remains  constant,  wliat  is  the  new  weight  of  steam  ?  What  is  the  heat  imparted  or  abstracted? 
What  amount  of  heat  should  be  imparted  to  a  mixture  of  steam  and  water  in  order  that  the 
pressure  and  temperature  may  rise  to  a  certain  point  ?  In  what  time  will  the  pressure  of  steam 
in  a  boiler  rise  by  a  certain  amount,  when  no  more  steam,  is  drawn  off  ?  If  in  two  vessels  com- 
municating by  a  cock,  we  have  mixtures  of  steam  and  water  of  given  pressure,  show  how  to  find 
the  condition  of  the  mixture  after  the  cock  is  opened. 


CHAPTEK  XX. 

THEOKY  OF  THE   CONDENSER. 

A.  Theory  of  the  Surface  Condenser. 


"When  the  piston  K  lias  reached  the  upper  end  of  its  stroke, 
the  slide  S  has  opened  the  port  o,  and  the  steam  escapes  into 
the  condenser  CI) ;  at  this  moment, 
therefore,  the  cylinder  steam  mixes 
with  the  low  steam  in  the  condenser. 
The  pressure  in  both  spaces  may  thus 
rise  to  double  the  ordinary  pressure 
in  CD.  We  can  calculate  it  as  in  the 
preceding  chapter.  But  now,  while 
the  piston  lingers  at  the  end  of  its 
stroke,  the  pressure  falls  by  reason  of 
the  cooling  effect  of  the  condenser. 
Heat  is  abstracted  under  constant  vol- 
ume, and  we  can  find  the  final  condi- 
tion, as  well  as  the  heat  abstracted, 
from  Equation  XXXY.  Then  the  pis- 
ton K  descends,  and  drives  the  steam 
in  the  cylinder,  now  at  the  condenser 
pressure,  under  that  pressure,  into  the 
condenser.  Here,  then,  we  have  mechanical  work  under  con- 
stant pressure.  This  work  is  transformed  into  heat,  and  this 
heat  is  also  absorbed  by  the  condenser. 

"We  see,  therefore,  that  in  every  stroke  there  are  three  stages : 
1st.  Mixture  of  cylinder  with  condenser  steam,  and  the  at- 
tainment of  a  mean  pressure. 

2d.  Lowering   of  this   pressure   by  cooling  under  constant 
volume. 

460 


THEORY  OF  THE  SURFACE  CONDENSER.  461 

3d.  Abstraction  of  heat  under  constant  temperature,  wliile 
tlie  low  cylinder  steam  is  forced  into  tlie  condenser. 

If,  therefore,  we  suppose  in  the  cylinder  AB,  before  opening 
of  the  port,  G  kilograms  of  steam  and  water,  we  have,  in  each 
kilogram  of  the  mixture,  x  kilograms  of  steam  and  1  —  x  kilo- 
grams of  water.  The  steam  quantity  in  AB  is  then  xG  kilo- 
grams, and  the  water  quantity  (1  —  x)  G  kilograms. 

If  Ave  had  only  pure  saturated  steam,  we  should  have  x  =  1, 
and  the  steam  quantity  in  the  cylinder  would  be  G  kilograms. 

Suppose  we  have  in  the  condenser  CD,  before  the  mingling, 
Gi  kilograms  of  steam  and  water,  and  in  every  kilogram  of  the 
mixture  X]_  kilograms  of  steam  and  1  —  x^  of  water.  The  steam 
weight  in  the  condenser  is  then  x^G^,  and  the  water  weight 
{^-x,)G,. 

As  soon,  now,  as  we  know  the  pressure  of  the  steam  in  the 
cylinder  and  in  the  condenser,  we  can  find,  according  to  the  pre- 
ceding chapter,  the  condition  of  the  steam  in  both  spaces  after 
the  port  is  ojDened.     We  have 


and 


Gxu  +  GiX^Ui 
■     G+G, 


gs  +  «2P2 


{q  4-  xp)  G  +  (qi  +  Xipi)  G^ 
G+  G, 


If,  as  before,  we  denote  the  fraction  on  the  right  in  the  first 
equation  by  ^Ji,  and  in  the  second  by  p,  we  have 

x^^u^  =  j9i,     or     0^3  -—  =^  5 
and 

Pa 
^2  +  --pi^P, 

and  thus,  by  a  few  trials  with  the  table,  can  find  q^  and  the 
pressure  ^h  and  the  temperature  fg  of  the  mixture  directly  the 
port  is  opened.  This  pressure  p^  is,  of  course,  greater  than  pi 
and  less  than  p)' 

Heat  now  is  abstracted  from  the  mixture,  until  the  pressure 
sinks  from  p^  to  pi.  In  order  to  find  this  heat  we  have  Equa- 
tion XXXV. 


462  THERMO  D  YNA  MICS. 

In  order  to  impart  to  1  kilogram  of  steam  and  water,  of  wliich 
X  kilograms  are  steam,  whose  temperature  is  t  and  pressure  p, 
a  pressure  pi  and  temperature  ^j,  we  must  add  a  quantity  of 
lieat  (or  subtract)  equal  to 

Q^  q-q^^  px  —  p^x, 
or 

«=.-..  +  »(M;). 

If  we  have  in  the  beginning  G  +  Gi  kilograms,  at  the  press- 
ure jjo  and  temperature  ^2)  the  steam  weight  per  kilogram  being 
a?2,  and,  by  abstracting  heat,  reduce  the  pressure  to  pi,  and  the 
temperature  to  t^,  then  we  have 

or 

Qi  =  (^2  +  x,p,)  {G+  Gi)-  (qi  +  xp,  ^){G+  Gi)' 

If,  as  on  Mie  preceding  page,  we  put 

(^2  +  a^apa)  {G  +  G^)  =  {q  +  xp)  G  +  (q^  +  x^p^)  Gi, 
and 

X2t(ii  {G  +  Gi)  —  Gxu  +  GiXiUi, 

we  have,  after  reducing 

The  difference denotes  the  excess  of  inner  latent 

u        Ui 

heat  of  1  cubic  meter  of  steam  in  the  cylinder  AB  before  the 
port  is  opened,  above  that  possessed  by  1  cubic  meter  of  steam 

in  the  condenser.     The  product  xu  (- ~)  G   is,  therefore, 

the  excess  of  the  inner  latent  heat  possessed  by  the  entire  steam 
volume  in  the  cylinder,  before  the  port  is  opened,  over  that  in 


THEORY  OF  THE  SURFACE  CONDENSER.  463 

tlie  condenser.  In  like  manner,  {q  —  ^i)  G  is  the  excess  of  sen- 
sible lieat  of  the  mixture  in  AB  over  that  in  the  condenser.  In 
other  words,  the  entire  expression  on  the  right  gives  the  ex- 
cess of  heat  in  the  mixture  in  AB  before  the  port  is  opened 
over  that  in  CD.  The  heat  quantity,  therefore,  which  must  be 
abstracted,  after  the  mingling,  from  the  entire  contents  of  both 
vessels,  in  order  that  the  pressure  7^2  may  sink  to  the  condenser 
pressure  p^,  is  equal  to  that  which  would  have  to  be  withdrawn 
from  the  contents  of  AB  before  the  mingling,  in  order  to  re- 
duce the  pressure  from  p)  to  jj^.  This  is,  in  fact,  evident,  for  if 
we  first  abstracted  this  heat  from  the  mixture  in  AB  and  then 
opened  communication  with  the  condenser,  the  final  condition 
would  remain  unchanged.  The  course  which  we  have  followed, 
however,  corresponds  perfectly  to  the  actual  changes  which 
take  place,  and  explains  why  we  have,  at  each  stroke  of  the 
piston,  sudden  variations  in  the  condenser  gauge. 

We  come  now  to  the  third  part  of  the  process,  the  abstrac- 
tion of  heat  under  constant  temperature,  while  the  now  low 
pressure  steam  in  the  cylinder  is  driven  out  under  constant 
pressure  into  the  condenser. 

The  volume  of  the  cylinder  AB  is 

G  (xu  +  a)  cubic  meters, 
or  if  0"  is  very  small  in  comparison  to  xu, 

Gxu. 

The  inner  latent  heat  in  every  cubic  meter  of  steam  after  the 
mingling  is  —  :  in  Gxu  cubic  meters  we  have  then 

Gxu^. 
u^ 

This  heat  must  be  abstracted.  If  we  suppose  the  piston  K 
to  have  an  area  of  1  square  meter,  the  distance  through  which 
it  must  go  is  Gxu  meters.  In  forcing  the  steam  then  under 
constant  pressure  ]Jx  into  the  condenser,  the  work  performed  is 
Gxupx,  and  this  corresponds  to  the  heat  A  Gxupi. 


464  THERMODYNAMICS. 

Hence  in  this  tliircl  period  we  must  abstract 

Q2  =  Gxu  —  +  GxuAiJ^  =  Gxu  (—  +  ^^1)  heat  units. 

The  total  heat  abstracted  is  then 

q=  Q^^Q^^\j-q^  +  xu  (^-^)]  G  +  xu  (-^^  +  Ap^  G 

=  \_q-q^  +  xu(^^^  +  Ap^~\^G.     .     .     (XLIV.) 

This  formula  can  be  simplified.  If  we  suj)pose  at  first  only 
pure  saturated  steam  without  water,  a;  =  1,  also  Ap^  is  very 
small  and  may  be  neglected,  and  then  we  have 

Q={q-qi  +  p)G. 

Since  q  +  p  is  the  "  steam  heat "  J,  we  have  also 

Q={J-qOG. 

For  low  temperatures  we  can  assume  that  qi  is  equal  to  the 
temperature.  If,  then,  instead  of  qi  we  insert  the  temperature 
of  the  condensed  steam,  or  generally  the  mean  temperature  of 
the  condenser,  ^1,  we  have 

Q  =  (J  —  ti)  G  heat  units. 

If  we  assume  that  we  require  for  condensing  the  steam,  n 
times  as  much  cold  water  as  steam,  and  if  the  temperature  of 
this  water  is  raised  from  to  to  ti  degrees,  where  f^  is  always  less 
than  tx,  then  the  heat  absorbed  by  the  water  is,  provided  that 
we  again  put  the  temperature  in  place  of  the  heat  of  the  liquid, 

nG{t,'-to). 

"We  have,  therefore, 

nG(h'-to)  =  (J-t,)G, 
or 

n  =  fr^ (XLV.) 

As  the  steam  heat  J  for  those  temperatures  which  occur  in 


THEORY  OF  THE  JET  CONDENSER.  465 

tlie  steam  engine  varies  but  little  (it  varies  liardly  23  heat 
units  between  100"  and  200""),  we  may  take  a  mean  value  for  it. 
Taking  then,  J  =  600,  we  have  the  practical  formula 

^^^600-f, ^^^^^^ 


EXAMPLE. 

How  much  more  water  than  steam  must  be  used  in  a  surface  condenser,  when 
the  water  enters  with  a  temperature  ^„  =  15"  and  departs  with  ^;'  =  35%  the 
mean  temperature  of  the  condenser  being  t^  =  46.2%  which  corresponds  to  a 
pressure  of  ^-  atmosphere  ? 

We  have 

600-46.2       553.8      ^^  „  ^. 

—  ~^^-  —  27.7  tnnes  as  much. 


35-15  20 

If  the  steam  used  per  stroke  is  0.15  kilogi-ams,  the  water  quantity  is 
0.15  X  27.7  =  4.155  Idlograms 
per  stroke,  or  since  1  kilogram  of  water  occupies  a  space  of  to^;u  cubic  meters, 
4.155  X  T-uVu  =  0.00416  cubic  meters. 

B.  Theoey  of  the  Jet  Condenser. 

Let  A  be  the  steam  cylinder  with  the  jDiston  KK.  The 
mixture  of  steam  and  water  in  it  weighs  G  kilograms.  In 
every  kilogram  of  this  mixture  there  are  x  kilograms  of 
steam  and  1  —  a;  of  water.  The  temperature  is  t,  and  the 
pressure  is^. 

B  is  the  condenser.  In  this  we  have  Gi  kilograms,  of  the 
pressure  p^  and  temperature  fi,  and  each  kilogram  contains  x^ 
kilograms  of  steam. 

Finally,  (7  is  a  vessel  filled  with  cold  water  at  the  tempera- 
ture (q.  It  holds  just  as  much  water  as  is  necessary  to  con- 
dense the  steam  used  per  stroke,  viz..  Go  kilograms.  Upon  the 
surface  of  the  water  is  a  piston  which  is  pressed  by  the  atmos- 
phere po- 

Both  A  and  0  communicate  with  the  condenser  by  cocks  a 
30 


466 


THEBMOD  TNAMICS. 


and  h. 
of   A 


Let  both  tliese  be  simultaneously  open.  Tlie  contents 
and  C  suddenly  mingle,  and  the  cold  water  injected 
reduces  the  pressure  in  both  A 
and  B  to  the  mean  pressure  of 
the  condenser  p^.  The  piston 
KK,  as  well  as  that  in  C,  now 
descends.  When  KK  reaches  the 
end  of  its  stroke,  so  has  the  pis- 
ton in  C,  and  all  the  water  has 
entered  the  condenser,  and  all 
the  steam  has  been  condensed. 
The  total  heat  in  A  is 


and  in  B, 


and  in  C, 


{q  +  xp)  Q, 


{qi  +  a^iPi)  Gi, 


qoGo- 


Hence  the  heat  in  all  three  vessels  is 


{q  +  xp)  G  +  {qi  +  a?aPi)  G^  +  q^G^. 

After  the  mixture,  all  the  steam  and  water  is  in  the  con- 
denser, and  we  have  there  a  mixture  of  steam  and  water  weigh- 
ing 

G  +  Gi  +  Gq  kilograms. 

In  each  kilogram  of  this  mixture  there  is  much  more  water, 
and  hence  much  less  steam,  than  before  a  was  opened  there 
was  in  A  and  B.  Let  the  steam  weight  in  each  kilogram  be 
£Ci',  then  Xi  is  less  than  x^,  and  we  have  in  the  G  -{■  Gi  +  Go 
kilograms 

Xi  {G  +  Gi  +  Go)  kilograms  of  steam, 


the  pressure  of  which  is  j^i  and  temperature  t^.  Since  the  heat 
of  the  liquid  is  qi,  and  the  inner  latent  heat  Pi,  the  heat  in  the 
condenser  is 

{qi  +  x,'p,)  {G  +  G^+  Go), 


THEORY  OF  TEE  JET  C0NDEN8EB.  467 

or,  denoting  G-  +  G^  +  Gohj  31, 

{qx  +  cci'a)  31  lieat  units. 

This  is  not  all  tlie  lieat  in  tlie  condenser. 

Wliile  tlie  piston  KK  moves  down,  it  overcomes  the  constant 
pressure  jh  through  a  certain  distance,  and  therefore  j)erforms 
mechanical  work  in  compressing  the  steam.  This  work  gen- 
erates heat.     Since  the  cylinder  volume  is 

{xu  +  a)  G  cubic  meters, 

the  work  performed  is 

{xu  +  (j)  Gpi, 
or,  neglecting  ff, 

xu  Gpx. 

The  heat  equivalent  to  this  work  is 

Apixu  G. 

Mechanical  work  is  also  performed  by  the  descent  of  the 
piston  in  C.  If  we  denote,  as  always,  the  volume  of  1  kilogram 
of  water  by  ff,  then  since  the  atmospheric  pressure  is  2^0^  ^^^ 
work  performed  is 

Go<ypQ, 
and  the  heat  equivalent  is 

A  Goffpo  heat  units. 
The  increase  of  heat  due  to  these  two  causes  is  therefore 

ApiXuG  +  AGo<^Po' 

If  we  add  this  heat  to  that  which  existed  before  the  mingling, 
in  the  three  vessels,  we  have  the  heat  in  the  condenser.   Hence, 

{q  +  xp)G  +  {q-i  +  iZiPi)  Gi  +  Goqo  +  A  ( Gjo^xu  +  Go(jpo)  =  {9.\  +  «i'Pi)  ^^' 


468  THERMODYNAMICS. 

Now,  as  a  matter  of  fact,  tlie  lieat  due  to  the  work  performed 
is  very  small,  so  tliat  we  can  still  have 

{q  +  xp)G  ^  (qi  +  x^Pi)  Gi  +  q^Go=  (q^  +  x^p^  M. 

Since  now  {qi  +  x^p^M—  qx{G  +  Gi+  G^  +  x^pxM,  we  have, 
after  reduction, 

{q  —  qx+  xp)  G  +  XyPx Gx  —  xIpxM  =  {q^  -  q^  Gq. 

Here  cc/  is  unknown,  but  it  can  be  easily  proved  that  x-^Pi  G^  = 
XiPxM. 

Before  the  mingling,  the  volume  v  of  1  kilogram  of  mixture 
in  the  condenser  was 

'G  —  x-iUx  +  <y, 

and  since  there  were  Gi  kilograms,  the  volume  ( Fi)  in  the  con- 
denser was 

Fi  =  Gx  {x{Ux  +  c). 

After  the  mingling,  the  volume  of  1  kilogram  is 

X  =  Xi'Ui  +  6. 

Since  the  pressure  and  temperature  are  the  same,  ih  is  the 
same.  But  after  the  mingling,  there  are  M  kilograms  of  steam 
and  water  in  the  condenser,  and  hence  the  volume  is 

Fi  =  M  {xlux  +  0-). 
Hence, 

Gx  {xxUx  +  c)  =  M{xxUx  +  a)  or  G^x^  —  Mxx* 

The  product  G-^Xx  is  the  steam  weight  in  the  condenser  at  the 
beginning,  and  Mxx  that  at  the  end.  As,  then,  this  steam  weight 
is  constant,  all  the  steam  in  the  cylinder  must  be  condensed. 
Since  now 

XxGx  =  XxM,  we  have  also 

^\PiGx  —  Xi'pxM, 

and  hence  the  equation  above  becomes 

(qx  +  qo)  GQ={q-~qx  +  xp)  G, 


THEOBY  OF  THE  JET  CONDENSER.  459 

We  have,  tlierefore,  tlie  water  required  for  condensation 

^       {q  +  xp-q,)G  (XLYIL)- 

If  we  make  liere  x  =  1,  and  put  for  q  +  p  tiie  steam  lieat  J, 
and  for  q^  and  qo  the  temperatures  fj  and  ^05  we  have 

or  if  Go  =  nG 

n=   J     .     .     . 


(XLYIII.) 


This  equation  differs  from  XLY.  in  that  t^  in  the  denomina- 
tor is  the  temperature  of  the  condenser  water,  while  in  XLV. 
f/  is  the  temperature  of  the  heated  condensing  water,  which  is 
always  less  than  that  of  the  water  of  condensation.  Since  here 
also  J=  600  about,  we  have 


600-^1 


(XLIX.) 


EXAMPLE. 

A  high  pressure  steam  engine  using  steam  of  5  atmospheres  has  a  condenser 
in  which  the  average  pressure  is  0. 1  atmosphere.  The  cooling  water  has  a  tem- 
perature of  18°.     How  much  more  water  than  steam  must  be  used  ? 

For  -jV  atmosphere,  t^  —  46°,  hence 

600-46      554     „.  ^.  , 

n  =  ^g  _  .^g-  =  gg-  =  20  tunes  as  much. 

If  then  the  steam  used  per  stroke  is  C.12  kilograms,  we  have  per  stroke 
20  X  0.12  =  2.40  kilograms  of  water  necessary,  or  j^  =  0.0024  cubic  meters. 

"We  have  already  noticed  the  fact  that  the  air  contained  in 
the  injection  water  is  set  free.  This  air  increases  the  press- 
ure in  the  condenser,  on  the  average  about  0.05  atmosphere. 
The  amount  of  injection  water  is  not  thereby  increased,  but  the 
air  pump  miist  have  greater  dimensions  in  order  that  it  may 
remove  both  air  and  water  from  the  condenser. 


CHAPTEE  XXL 

THE  FLOW  OF  STEAM  AND   HOT  WATER  THROUGH  ORIFICES. 

A.  Flow  of  Steam  through  Orifices. 

As  in  Chapter  X.,  Part  L,  let  ABCD  be  a  large  vessel  with,  a 
narrow  discharge  pipe  EF.  In  the  first  is  a  piston  HJ,  of  F  sq. 
meters  cross-section,  and  in  the  pipe  a  smaller  one  G,  of/  sq. 

meters.     Upon  HJ  we 

JS g;g  have  the  pressure  of  ^ 

kilograms   per  square 

„  ^,,^^;^.^  meter.      The  pressure 

-l-B  u'rl  >'^H  ?  P<<r      /    upon  G  is  ^1  kilograms 


D 


per  square  meter,  and 
Px  <  P-  Suppose  the 
space  between  the  pis- 
tons filled  with  some 
liquid,  as  water.  Let  the  piston  HJ  move  through  the  distance 
s  per  second,  and  G  through  s^.  Then  the  work  of  the  first  pis- 
ton is  Fsp,  and  of  the  second /si^j. 

The  force  Fjj  has  to  perform  two  works.  First,  it  must  over- 
come the  constant  resistance  f2Ji,  with  uniform  velocity,  and 
second,  it  has  to  impart  to  every  water  particle  which  enters 
the  pipe  an  increased  velocity,  so  that  the  velocity  s  becomes 
•S'l.  Since  the  volume  fsi  issues  through  F  per  second,  the 
weight  of  this  volume  is,  if  1  cubic  meter  weighs  y  kilograms, 
fsiy.  To  increase  the  velocity  of  this  from  s  to  s^,  requires  the 
work 

„2  ^ 

-Ar- 


The  force  Fp,  which  drives  the  piston  HJ,  has  then  to  per- 

470 


FLOW  OF  STEAM  THBOVGH  0BIFICE8.  471 


form  not  only  tlie  work  fsi^ih,  but  also  tlie  work  -^—^ fsiy 

Hence 

Fsp  ^fs^2h  +  -^-^^A?"' 


Fsp  -fs^jh  =  '  2„     A^ (^• 


Since,  also,  Fs  =fsi,  we  liave 


p-lh  =  -^^y (LI.) 

This  is  a  well-known  formula  of  hydraulics  of  frequent  ap- 
plication. If  we  assume  the  vessel  ABCD  very  great,  with 
reference  to  the  pipe  EF,  or  the  diameter  very  great  in  com- 
parison to  that  of  the  orifice  E,  the  distance  s  is  very  small  in 
comparison  to  Sj,  and  we  have 

or  putting  v^  in  place  of  s^, 

p-Pi  =  '^^r (Lli.) 

We  omit,  of  course,  here  the  fact  that  the  pressure  of  the 
water  particles  above  the  orifice,  due  to  their  own  weight, 
assists  the  force  Fp. 

Now  let  us  sup230se  that  instead  of  water  between  the  piston 
HJ  and  G,  we  have  saturated  steam.  Let  the  pressure  of  this 
steam  in  ABCD  be  p.  The  piston  HJ  is  pressed,  as  before, 
from  left  toward  the  right  by  the  force  p.  If  here  also  the 
pressure  p^  upon  the  piston  G,  in  the  pipe  EF,  is  less  than  p, 
the  efflux  of  the  steam  is  essentially  different  from  that  of 
water,  and  is  completely  similar  to  that  of  a  gas.  The  steam 
expands  when  it  is  subjected,  on  one  side,  to  a  less  pressure 
than  on  the  other.  For  this  reason,  one  kilogram  of  steam  in 
the  pipe  EF  occupies  a  greater  space  than  a  kilogram  mABCD, 
that  is,  the  specific  volume  v  in  EF  is  different  from  that  in 


472  THEBM0DTNAMIG8. 

ABCD.  Tlie  distance  between  the  molecules  when  they  pass 
the  orifice  E  is  greater,  and  they  have  then  been  moved  apart. 
This  motion  can  be  caused  by  heat  received  from  without,  or 
by  the  heat  of  the  steam  itself.  In  the  latter  case,  a  part  of  the 
molecular  motion  of  the  steam  is  converted  into  outer  motion. 
In  the  case  of  permanent  gases,  the  pressure  pi  and  the  spe- 
cific volume  Vi  in  the  pipe  can  be  different  from  that  in  the 
vessel,  while  still  the  temperature  is  unchanged  (Chaj).  X., 
Part  I.)  With  steam  this  cannot  be,  so  long  at  least  as  it  is  in 
the  saturated  condition.  For  this  condition,  so  long  as  the 
temperature  remains  constant,  the  pressure  is  unchanged. 

Let  us  conceive  that  the  expansion  of  the  steam  while  pass- 
ing through  E  extends  to  *some  distance,  as  shown  by  the 
dotted  lines.  As  the  steam  molecules  reach  the  first  arc,  their 
expansion,  or  the  increase  of  their  mutual  distances,  is  still 
small.  As  they  approach  the  innermost  arc,  it  is  greater,  until 
in  the  pipe  EF  the  volume  is  Vi.  Since,  then,  the  mutual  dis- 
tance of  the  molecules  increases  gradually,  the  pressure  exerted 
by  any  molecule  at  any  instant  upon  the  next  one  which  lies 
nearer  the  orifice  E  is  greater  tham  the  counter-pressure  of 
this  last  molecule  only  by  an  infinitely  small  amount.  If,  then, 
the  expansion  follows  any  given  law,  we  can  calculate  the  work 
necessary  for  this  expansion. 

If,  for  example,  we  suppose  the  expansion  to  follow  the  law 
of  constant  steam  weight,  heat  must  be  imparted.  We  havp 
seen  (page  405)  that  the  heat  required  by  1  kilogram  of  satu- 
rated steam,  expanding  according  to  this  curve,  varies  with  the 
temperature  ;  for  high  temperatures  it  is  less  and  for  low  tem- 
peratures greater.  If  the  steam  pressure,  for  example,  in 
ABCD  is  5  atmospheres,  or  ^9  =  5  x  10334  kilograms,  and  the 
pressure  on  G  per  square  meter  is  1  x  10334  kilograms,  we 
have  from  Table  III.  for  the  heat  imparted  per  kilogram  during 
expansion, 

200.46  -  148.47  =  51.99  heat  units. 


(Compare  the  example,  page  412.)     The  temperature  falls  from 
152.2 ^  corresponding  to  5  atmospheres,  to  100°. 

If  we  denote  generally  the  heat  which  must  be  imparted, 
when  the  expansion  of  the  steam  at  efflux  follows  the  curve  of 


FLOW  OF  STEAM  THBOUOH  0BIFICE8.  473 

constant  steam  weight,  by  P,  tlien  tlie  work  corres]ponding  is 

P  .        . 

-j^.     It  is  this  work,  which  in  the  efflux  of  steam  (just  as  for 

gas)  is  to  be  added  to  the  work  Fsp  (Equation  L.),  and  which 
in  combination  with  this  work  generates  the  velocity  lo  and 
overcomes  the  constant  pressure  75.  in  the  pipe  PJF  with  this 
velocity.  If  we  assume  that  1  kilogram  of  steam  issues  per 
second,  and  denote  the  volume  of  this  kilogram  when  in  the 
vessel  ABCD,  by  v,  and  when  in  the  pipe  EF,  by  Vi,  we  have 
in  Equation  L.,  Fs  =  v  and/sj  =  v^,  and  have  then  for  the  work 
necessary  to  impart  the  velocity  to  to  each  kilogram 

P  '  tv"   ^ 

since  in  Equation  L.,/siX  can  be  put  =  1. 
The  specific  volume  v  or  v^  is  given  by 

V  =  (xtt  +  c) 
Vi  =  (xui  +  g), 

because  the  steam  weight  x  is  constant.     For  x  =^\ 

V  =  u  +  (J 

Vx  =  111  +   ^• 

It  is  easy  to  see  how,  from  the  pressures  p  and^i,  by  the  aid 
of  the  above  equations,  the  velocity  tv  can  be  found.  In  prac- 
tice, the  case  here  considered  occurs  but  seldom  if  at  all,  and 
we  shall  not,  therefore,  pursue  it  further. 

Velocity  of  Efflux  when  Heat  is  neither  Added  nor  Abstracted. — 
The  efflux  of  steam  from  the  safety-valve  of  a  boiler,  or  from 
the  cylinder  of  an  engine,  takes  place  without  the  addition  of 
heat  from  without.  Let  us  find  then  the  velocity  iv  under  the 
assumption  that  heat  is  neither  added  nor  abstracted  by  outer 
bodies.  The  expansion  then  follows  the  law  of  the  adiabatic 
curve,  and  by  the  gradual  change  of  pressure  from  ^j  to  pi,  the 
work  of  the  molecules  is  (Equation  XXYII.) 


A 


qt  +  xp  -  XiPi 
A 


474  THEBMOBYNAMIGS, 

Hence  we  liave 

p^^p^^^  +  l — i^___/ —       '^J  '  ^     '^ 

Multiplying  botli  sides  by  A,  and  putting  v  =  xu  +  ff,  and 

q  +  xp  +  Ajjux  —  (q^  +  XiPi  +  AjhXiUi)  +  Act  {2^  ~ 2h) 

=  a|.    .    (LIT.) 

Here  we  can  disregard  the  last  member  on  tbe  left.  Also 
remembering  tliat  x  {p  +  A2Ju)  =  xr,  and  x^  (pi  +  ^pi^i)  =  x^r^, 
we  have 


2V 


^^  =  <I-(li+  ^r-  x^i\    .     .     .      (LV.) 

Since  p,p\  and  x  are  known,  q,  q-^,  r  and  i\  are  also  known,  and 
x-^  can  be  found  from  Equation  XXV., 


^  +  "jt"  —  ^1  +   —jT 

whence 

xr\        r-i 


^1=  i^-n+  ^y  -^. 


T  .  . 

The  values  of  r  and  -=  are  given  in  Table  III.     If  then  x^  is 


thus  determined,  we  have 


|/|(^-«^ 


^^=i/  ^{^-^1  +  ^^^  -  ^in) ' 


Since  2^7=  2  x  9.81  meters, and  -  =  424, 


IV  =  91.2  Vq  —  qi  +  icr  —  x^r^      0     .     (LVI.) 
Let  us  illustrate  the  above  by  an  example. 


FLOW  OF  STEAM  THBOTIGH  0BIFICE8.  475 


Steam  issues  from  the  safety-A'alve  of  a  steam  boiler,  in  wliich  the  pressure  is 
5  atmospheres,  into  the  air.  What  is  the  velocity  of  elflus  iv,  when  we  assume 
the  efflux  to  take  place  without  friction,  and  that  only  pure  saturated  steam, 
without  admixture  of  water,  approaches  the  orifice  ?  How  much  steam  is  con- 
densed during  efflux  ? 

We  have  evidently  to  answer  the  last  question  first.     From  Table  III., 


•J=  1.174,     |p  =  1.438,     r  =  0.447,     r,  =  0.314, 

hence 

Xi  =  (0.447  -  0.314  +  1.174)  --  1.438  =  0.908  kilogram. 

Hence  during  efflux,    1  —  0.908  =  0.092  kilograms  of  steam  are  condensed. 
Further, 

q  =  153.74,     q^  =  100.5,     r  =  499.19,     r,  =  536.5, 
hence 

q-qi  +xr-  x^7\  =  153.74  -  100.5  +  499.19  -  0.908  x  536.5  =  65.29, 
and 

w  =  91.3  |/64.29  =  91.2  x  8.08  =  736.9  meters. 


Transformation  of  tlie  Preceding  Equations. — The  preceding 
formulse  can  only  be  used  when  we  have  our  tables  at  hand. 
Let  us  see  if  we  can  find  iv  without  them. 

First,  we  can  put  for  q  and  q^  the  temperatures  t  and  t^.  Thus 
if  we  assume  the  specific  heat  of  water  as  constant,  and  take  it 
at  1.0224,  which  corresponds  to  about  145°,  we  have 


Also, 


and 


hence 


q-qx=  1.0224  {t  -  t,)  =  1.0224  (2^  -  T^^ 


T  T 

r  =  c  loer  nat  —  =  1.022  loar  nat  — 

^  a  °  a 


T  T 

r^=c  log  nat  --  =  1.022  log  nat  --  , 
a  a 

-r,  =  l  022  log  nat  ^  =  2.353  log  ~ . 


Further,  according  to  Clausius  (page  379), 

r  =  607  -  0.708^,         r^  =  607  -  0.708fi. 


476  THEBMODTNAMIGS. 

Hence 

foo/^oi      T   ,  «(607- 0.70801      607-0.708^1 
X,  =  ^2.353  log^  +  -^ ^ -^  J -^ i- . 

Equation  LVI.  becomes 

w  =  91.2  Vl.022  it  -t,)  +  x  (607  -  0.7080  -  x,  (607  -  0.708^0 

(LVII.) 

Let  us  calculate  from  these  last  two  formulas  oc^  and  w  for 
tlie  last  example. 

We  liave  ^i  =  100,     and     Tt  =  273  +  100  =  373  ;  hence, 

607  -  0.708^1       607  -  70.8      ,  ,^^ 

T, =  ^    373       -  ^•^^^- 

Further, 

2.353  log|J  =  2.353  log^^^?^^^  =  2-353  x  0.0569  =  0.1339. 

-L  I  O  I O 

Hence 

g;  (607 -0.7080  ^  607  -  0.708  x  152.2  _-,-,,,. 
T  ~         273  +  152.2    --^-L'^- 

We  have  then 

X,  =  (0.1339  +  1.174)  -  1.438  =  0.909  kilogTams. 

By  means  of  the  tables  we  found  before  x^  —  0.908. 
We  have  now 

to  =  91.2^1.022  X  52.2  +  499.24  -  0.909  (607  -  70.8) 


91.2  V53.35  +  499.24  -  487.41 


=  91.2a/65.18  =  91.2  x  8.07  =  735.98  meters. 

This  value  agrees  closely  with  that  found  before. 

If  the  pressures  p  and  p^,  or  the  corresponding  temperatures 
differ  but  little,  we  can  put 

rp  rp rp 

2.353  log -J^=^^^^ 


^.  =  i^^  +  y;x,. 


FLOW  OF  STEAM  THROUGH  0BIFIGE8.  47( 

We  slioulcl  have  then 

T  -  T^       xr\      Ti 

^  X  ■ 

If  we  insert  this  in  LVI.,  and  remember  that  T  —  Ti  =  t  —t^, 
and  that  we  have  without  great  error  c{t  —  t^  =t  —  t^,  we  have 

«(;=:91.2j/^(r-ri)   .     •     •     (LYIII.) 
or  finally, 


,^91,,/^(607^-0.708T^^_    ,     (^jX^j 


With  what  velocity  does  the  steam  issue  from  the  boiler  of  a  low-pressure  en- 
gine, the  pressure  being  only  1.2  atmospheres,  when  x  =  1? 
In  this  ease  t  =  105.2°  and  t  =  100, 


i/607- 0.708  X  105.: 

y     9.17.Q 


w  =  91.2  r 273  +  105  2    '     ^  ^"^  ^  ^^^"^^  meters. 


Anotlier  Expression  for  tJw  Velocity  of  Efflux. — ^We  have  made 
use  above,  page  473,  of  the  expression 

A -J 

for  the  work  of  1  kilogram  of  steam  in  expanding  from  the 
pressure  p  to  p-^.  On  page  436,  we  have  seen  that  this  work  is 
given  by  the  expression 


^-i^t-e)"] 


where  the  value  of  m  depends  not  only  uj)on  p  and  j9i  but  also 
upon  the  proportion  of  water  in  the  mixture.     If  we  have  only 


478  THERMODYNAMICS. 

pure  saturated  steam  without  water,  m  lias  the   mean  value 
1.135. 

We  have  then 

m-l 
11? 


,„-,,,,_J^[i_(M»J.    .    (,^, 


2^      ^^"      ^^^^^  '  m  -  1  L        \io 

The  law  of  the  adiabatic  curve,  that  is,  the  law  of  expansion 
of  the  steam  during  efflux,  is 

hence 

Vi=  f^jv    and    piVi=:j)i.     "^ P '" ^• 
Inserting  this  value  oijJiVi  and  reducing, 

m  -1 


/  m  -  1 

=  4.43i/j^[l-(f)»].    (LXII 


P' 

We  can  now  find  by  this  formula  the  velocity  w  for  the  ex- 
ample on  page  474 

First,  -^^=  0.2.    Put  m  =  1.135,  then  (^^  "  =(0.2)'""! 
By  the  aid  of  logarithms  we  find  this  equal  to  0.8257.     Hence 

m-l 

Tl  -  (^)  "   1=1-  0.8257  =  0.1743.     Further,  for  5  atmo- 
spheres V  =  xu  +  <j  =:  0.364.     Since  p  ==  5  x  10334, 

»  =  JASiA-''''  -  '  -  l°f  ^  X  "-^"^  X  0.1743"^  733.38  m.- 
^  0.135 

ters. 

This  result  agrees  closely  with  that  already  found. 


FLOW  OF  STEAM  THBOUOH  ORIFICES.  479 
For  practical  use,  Eq.  LXII.  can  be  sii-iplified.     Thus,  mak- 
ing m  —  1.135,  and  taking  tlie  sq.  root  of :.  10334,  we  have 


/  m  — 1 

=  1305.83  j/J?v[l-(B^   '"  J.     (LXIII. 


Here  ^  is  expressed  in  atmospheres. 

If  the  steam  contains  10  per  cent,  of  water,  m  —  1.125,  and 

/  TO-  1 

w  =  1351  j/  i)z;  [1  -  ('^)    "  1  .     (LXIV.) 
Table  IV.  gives  for  different  pressure  ratios  ( —  ]  the  values 

m  —  1  1 

^— j        and  off— j     ,  which  last  serves  to  determine 
the  steam  volume  and  weight  which  issues  per  second. 

Steam  Volume  and  Weight  per  Second. — We  have  thus  far 
assumed,  for  the  sake  of  simplicity,  that  one  kilogram  per  sec- 
ond passes  through  E,  and  have  found  lu  under  this  assump- 
tion. The  velocity  evidently  will  not  change  when  2,  3,  4,  or 
G  kilograms  issue  per  second.  Apart  from  the  pressures  p  and 
p]^,  this  quantity  depends  only  upon  the  area  of  the  orifice.  If 
this  area  is  F,  the  volume  issuing  per  second  is 

Ti  -  Fio, 

provided  that  there  is  no  contraction  of  the  stream,  as  is  the 
case  with  water  and  other  liquids.    If  G  is  the  weight  of  steam, 


or  since 


1 


Vx  =  (  —  )   vG  cubic  meters. 


480  TEEBM0DTNAMIC8. 

Hence 


Fw 


and 


■p  Y"  1 


(fX^-' 


G  =  Fiv  C^ j  " .  ^  kilograms.     .     .     (LXV.) 
In  order  to  show  the  use  of  Table  IV.,  we  take  an  example. 


EXAMPLE. 

Let^  =  4,  and  js,  =1  atmosphere.      The  steam  at  4  atmospheres  is  pure 
saturated,  without  water.     What  is  u  and  G  ? 

We  have  ^  =  0.25.     We  find  in  the  column  for  —  the  number  0.25.    In  the 
P  P 

/»,\  ~^ 
same  horizontal  line  we  find  for  the  column  giving  1  —  (  .T  )         >  ^^^  number 

0.1521.     Hence 

w  =  1305.83  V4.X  V  X  0.1521 . 

Since  v  =  xu  +  d,  and  u,  for  4  atmospheres,  is  from  Table  II.,  0.447,  we  have 
V  =z  0.447  +  0.001  =  0.448.     Hence 


=  1305.83  V4:  X  0.448  x  0,1521  =  681.9  meters. 


Again,  for  0.25,  we  find  in  the  column  for  (  i^  )       (m  being  1.125)  0. 

\P  J 


Hence 


&  =  681.9  X  0.2948  x  ^^^to  ^  448.31^  kilograms. 
U.44o 


Calculation  of  the  Size  of  Safety-  Valve. — In  general,  we  make 
the  area  F  of  the  valve  so  great  that  it  will  discharge  10  to  20 
times  as  much  steam  in  a  given  time  as  the  boiler  can  generate. 
This  latter  quantity  depends,  however,  upon  the  heating  sur- 
face, hence  the  valve  and  heating  area  must  stand  in  a  certain 
relation.  According  to  Prussian  regulations  (Weisbach,  Vol.  II., 
Art.  434),  the  valve  area  should  be  -^xhn  o^  the  heating  area. 
This  gives  indeed,  for  high  pressure  boilers,  a  greater  safety 


FLOW  OF  STEAM  THBOUGH  0BIFIGE8.  481 

than  for  low  pressure,  since  for  tlie  latter  tlie  steam  issues  with 
less  velocity. 

We  shall  now  show  by  an  example  how  the  diameter  of  the 
valve  (d)  may  be  found  under  the  assumption  that  the  steam 
issues  with  the  velocity  given  by  our  formula,  and  that  the  loss 
of  velocity  due  to  friction,  etc.,  is  little  or  nothing. 

EXAMPLE. 

What  diameter  {cl)  should  the  safety-valve  of  a  steam  boiler  have,  which  gen- 
erates per  hour  250  kilograms  of  steam  of  4  atmospheres,  for  30-fold  security  ? 

We  have  already  found,  in  the  last  example,  the  weight  of  steam  issuing  per 
second  through  the  area  F  under  a  pressure  of  4  atmospheres.     It  was 

G  =  M&.ZF. 

If  the  diameter  of  the  valve  is  d,  we  have 

O  =  448.3  ^^  =  448.3  x  0.785^2. 
4 

The  boiler  generates  350  kilograms  of  steam  per  hour,  or  ■:^^§(s  =  0.0695  kilo- 
grams per  second.  Since  we  wish  30-fold  security,  we  have  for  G,  20  x  0.0695  = 
1.39  kilograms.     Hence 

1.39  :=  448.3  X  0.785d-^,    ov    d  ^  0.067  meters  =  6.7  centimeters. 


B.  Efflux  of  Hot  Water. 

In  the  chapter  upon  the  adiabatic  curve  of  steam  and  liquid 
mixtures,  we  have  seen  that  steam  is  generated  when  a  mixture 
of  water  and  steam  containing  a  preponderance  of  the  first  ex- 
pands adiabatically.  When,  then,  hot  water  flows  from  the 
vessel  ABGD,  the  particles  expand  in  approaching  the  orifice 
F,  and  steam  is  formed.  If,  then,  we  have  simply  hot  water 
and  no  steam,  the  Equation  XXV.,' page  423, 


xr  Xir]_ 


becomes,  since  x  =  0, 


31 


482  THERMODYNAMICS. 

Since  r  is  always  greater  tlian  r^  so  long  as  the  temperature 
of  the  water  is  greater  than  that  in  the  pipe  EF,  -^  is  posi- 

tive,  and  since  x^  is  the  steam  quantity  at  the  orifice  E,  we  see 
from  the  equation,  that  by  the  efflux  of  hot  water  steam  is 
generated,  and  that,  therefore,  both  steam  and  water  issue. 
The  weight  of  this  steam  in  every  kilogram  of  the  mixture  is 

cci  =  (r  —  Tj)  X  — i . 

If  we  put  this  value  of  ajj  in  Equation  LV.,  page  474,  we  have 

A'^^  =  q-q,-{r-r,)n     .     .     (LXYL) 


w  =  91.2  Vq-qi-{r-  r,)  T,  .     (LXYII.) 

If  instead  of  1  kilogram,  G  kilograms  issue  per  second,  and 
if  F  is  the  area  of  the  orifice,  we  have  for  the  steam  weight 
per  second 

D  =  XiG  kilograms, 
and  the  water  weight 

W=  ().  —  x^  G  kilograms. 

The  volume  of  1  kilogram  of  the  mixture  at  the  orifice  is 

Xxlh  +  <^i 

hence  of  G  kilograms  it  is 

{x^Ui  +  g)0-  cubic  meters. 
We  have  therefore 

{XiUi  +  ())  G  =  Fw, 


and 


^-^£h-  ■  ■  ■  (i^^™-> 


FLOW  OF  STEAM  THROUGH  0BIFICE8,  483 


EXAMPLE. 

Hot  water  flows  from  the  test  cock  of  a  boiler  under  the  pressure  of  5  atmo- 
spheres. What  is  the  specific  steam  weight  at  the  orifice  ?  With  what  velocity 
to  does  the  mixture  issue  ?  What  is  the  discharge  G  per  second  ?  How  much 
steam  and  water  are  contained  in  the  mixture  ? 

For  5  atmospheres  r  =  0.447,  for  1  atmosphere  Tj  —  0.314,  and  -^  =  1.438. 

Ti 
Hence  x^  =  (0.447  -  0.314)  h-  1.488  =  0.093  kilograms.   We  have,  then,  as  much 
steam  as  in  the  efflux  of  saturated  steam  alone  is  condensed  (page  475). 

Since,  further,  q  =  153.74,     q^  =  100.5,  we  have 


w  =  91.2  4/153.74  -  100.5  -  (0.447  -  0.314) 373 
:^  91.2  X  1.9  =  173.28  meters. 

For  1  atmosphere  w^  =  1.650,    hence  x^u^  +6  =0.092  x  1,650  +  0.001 
=  0.154,  and 

^  =  S^  F^  1125  kilograms. 
0.154  ° 

•  he  steam  weight  issuing  per  second  is 

D  =x^G  =  0m2  X  1125i^=  103.5^ kilograms. 

The  water  weight  is  therefore  (1125  —  103.5)  F  =  1021. 5i^.  If  then  F  is  given, 
we  can  find  G,  D  and  W.  If,  for  example,  i^  is  1  square  centimeter  =  i otttto  . 
=  0.0001  square  meter,  we  have  (r  =  0.1125  kilogram,  Z>  =  0.01035  and  If 
=  0.10215  kilogram  per  second.  The  volume  of  water  is,  since  1  kilogram 
=  0.001  cubic  meter,  0.1125  x  0.001  =  0.0001125  cubic  meter. 

We  see  from  this  calculation,  liow  to  find  for  any  time  the 
quantity  of  hot  water  and  steam  which  flows  through  a  given 
orifice  under  a  certain  pressure.  Zeuner  was  the  first  to  en- 
deavor to  determine  how  far  the  theoretical  result  agrees  with 
the  fact.  "I  allowed,"  he  says  in  his  Warme  Theorie,  "water 
to  flow  from  a  locomotive  boiler  under  6  atmospheres  pressure, 
and  endeavored  to  collect  and  measure  the  issuing  water.  In 
spite  of  variations  in  the  method  of  experiment,  the  measure- 
ment did  not  succeed.  The  steam  carried  off  with  it  the 
greatest  part  of  the  water."  There  remains  only  in  further 
experiments  of  this  kind,  to  measure  therefore  the  efflux  in 
the  boiler  itself. 


484  THERMODYNAMICS. 

The  following  table  gives  for  different  pressures  the  corre- 
sponding Telocity  of  efflux,  the  specific  steam  quantity  at  the 
orifice,  and  the  discharge  in  kilograms  per  second,  according 
to  Zeuner. 


Pressure 

in 
atmosp. 

Velocity 

of 
efflux. 

Specific  steam 

weight  at 

orifice. 

Efflux  G  in 
kilogr?.  per  second 
for  cross-section  F. 

10. 

Xx. 

G. 

2 

69.53 

0.038 

1095  F 

3 

112.85 

0.062 

1102  " 

4 

145.07 

0.079 

1106  " 

5 

171.02 

0.093 

1109  " 

6 

192.98 

0.105 

1113  " 

7 

212.00 

0.115 

1116  " 

8 

228.87 

0.124 

1118  " 

9 

244.14 

0.132 

1120  " 

10 

258.02 

0.139 

1123  " 

11 

270.78 

0.145 

1124  " 

12 

282.64 

0.152 

1127  " 

13 

293.71 

0.157 

1128  " 

14 

304.09 

0.163 

1130  " 

This  table  shows  the  remarkable  result  that  the  discharge  is 
nearly  constant  whether  the  pressure  be  small  or  great.  The 
specific  steam  weight  a^i,  on  the  other  hand,  increases  with  the 
pressure,  a  proof  that  the  total  steam  weight  issuing  increases 
with  the  pressure.  Since,  then,  the  entire  discharge  is  nearly 
constant,  the  water  weight  issuing  per  second  must  decrease 
with  increasing  pressure.  It  follows,  that  for  the  same  con- 
tents and  area  of  orifice,  a  low  pressure  boiler  can  be  emptied 
in  about  the  same  time  as  a  high,  provided  that  in  both  the 
initial  pressure  is  constant  during  the  efflux.  If,  however,  dur- 
ing efflux,  the  pressure  gradually  sinks,  the  weight  of  the 
issuing  mixture  indeed  remains  about  the  same,  but  the  steam 
weight  decreases  while  the  water  weight  increases.  We  have 
then  almost  the  inverse  phenomena  from  the  efflux  of  cold 
water. 

In  order  to  make  apparent  the  difference  between  the  efflux 
of  cold  and  hot  water,  we  give  the  following  small  table.  The 
velocity  of  efflux  is  found  from  page  471, 


p-p^  =  ^y 


FLOW  OF  STEAM  THROUGH  ORIFICES.  485 

where  7,  the  weight  of  one  cubic  meter  of  water,  is  1000  kilo- 
grams. Since  c  is  the  volume  of  1  kilogram,  y(j  i&  the  volume 
of  1000,  or  of  1  cubic  meter.     Hence 

1 
and    i(?  =  a  (]j  —  p^  2g    or 


2v  =  4:A3V(y{2)-2h)     •     .     .     (LXIX.) 

If,  for  example,  water  flows  under  the  constant  pressure  of  4 
atmospheres  from  a  vessel  into  the  air,  the  velocity  of  efflux  w 
is,  neglecting  all  resistances. 


w  =  4.43  V 10334  x  ^ ^^  ir  (4=  -  1)  =  4.43  V10334  x  3 
=:  24.66  meters. 

The  water  volume  per  second  is  then  Fw,  and  the  water 
weight 

G  =  10002^i^  =  24660i^  kilograms     .     (LXX.) 

Hence  we  have 


'ressnre. 
P- 

4 

8 

Velocit3^ 

24.66 

37.67 

Efflux. 
G. 

24660jP 
37673i^ 

12 

47.23 

47226i^ 

"While,  therefore,  for  the  same  pressure  the  velocity  of  the 
issuing  water  is  much  less  than  for  a  mixture  of  steam  and 
water,  the  discharge  is  much  greater.  The  explanation  is  as 
follows :  The  steam  weight,  in  spite  of  its  great  velocity,  is 
very  small,  but  even  this  small  weight  occupies  a  relatively 
large  space,  and  fills,  in  part,  the  orifice  so  that  the  water  quan- 
tit;^  is  small. 

It  is  much  to  be  desired  that  these  theoretical  investigations 
may  be  tested  by  thorough  and  exact  experiments. 

Cases  in  ivMcli  the  Hot  Water  Issues  loitli  tJie  same  Velocity  as 


486  THEBMODYNAMIGS. 

tlie  Cold. — The  complete  equation  for  the  velocity  of  the  issuing 
steam  was,  from  Equation  LIV.,  page  474, 

q  +  xf)  +  Apux  —  {qi  +  x^pi  +  Ap^x^u-^  +  Aa {p  —  p^  =  A^ , 

^9 


10^ 

q  +  xr-q^-  x^r^  +  Aff  {p  -  p^)  =  A  ^  . 

If,  now,  instead  of  steam  we  have  only  hot  water,  x  =  0,  and 
we  have 

q-qi-x,n+Aff{p-2h)  =  A^, 
Now,  from  page  474, 

and  since  here  also  a?  =  0, 

T 

x^={r  -  Ti)  --i ,     or     x,i\  =  (7  -  n)  T^' 

Instead  oi  q  —  q^  we  can  put  c{t  —  ti)  =  g(T  —  T^  and  have 


c{T  -  T,)  -  (r  -  r,)  T,  +  A<y  {p  -p,)  =A^^.     (1.) 


If  the  pressures  j^  and  p^,  or  the  temperatures  T  and  T^  are 
but  little  different  from  each  other,  we  have 

2.3026  lor       -  ^ 


From  page  475 

r  -  r,  =  2.3026c  log -^^  =  c  ?^V 


FLOW  OF  STEAM  THROUGH  ORIFICES.  487 

In  equation  (1)    tlie  two  first  members  tlien  are  equal,  and 
we  have 


i(? 


=  a{p-p,)  =  ^-^.    ,    .     (LXXLj 


This  is  the  same  equation  which  we  have  found,  page  471, 
for  the  efflux  of  water  under  ordinary  circumstances.  We  have, 
however,  called  attention  there  to  the  fact  that  the  pressure 
due  to  the  head  of  water  over  the  orifice  should  be  taken  into 
account  in  finding  the  velocity  of  efflux.  The  same  is  the  case 
for  hot  water.  Since  y  is  the  weight  of  one  cubic  meter  and  p 
the  pressure  in  kilograms  per  square  meter,  we  have,  when  p  is 
expressed  in  atmospheres,  for  the  height  of  a  column  of  water 
which  would  exert  this  pressure, 

10334P  ^  10334  _ 

y  lUUu 

If  the  head  of  water  is  Ji  meters  above  the  center  of  the 

orifice,  we  have  instead  of  ^ ,  ^  +  h.     Hence  the  theoretical 

y    V 
velocity  of  the  cold  water  is 


U3,|/ 


ji  +  l — ^.  .    .    (LXXII.) 


and  this  will  be  the  velocity  also  of  the  hot  water  when  p  and 
Pi  are  nearly  equal. 

The  efflux  of  hot  and  cold  water  must  then  be  the  same,  if 
steam  formation  during  efflux  is  prevented.  This  can  only  be 
the  case  when  we  abstract  from  the  water  as  much  heat  as  the 
steam  requires  for  its  formation.  If  the  heat  of  the  liquid  in 
the  vessel  is  q,  and  outside  q-i,  we  must  abstract  for  each  kilo- 
gram of  water  at  the  orifice  the  heat  q  —  q^.  When  this  is  the 
case,  the  formula  for  efflux  of  cold  water  will  apply  also  to  hot 


488  THERM0DTNAM1C8. 

water,  because  the  diminution  of  volume  of  tlie  water  in  pass- 
ing from  tlie  higher  to  the  lower  temperature,  which  in  the 
efflux  of  gas  and  steam  is  so  great,  is  for  water  so  small  as  to 
have  no  influence  upon  the  flow.  The  velocity  iv,  and  the  dis- 
charge in  kilograms  per  second  or  minute,  must  then  be  the 
same,  in  this  case,  both  for  cold  and  hot  water. 


CHAPTEE  XXn. 


CONSTEUCTIONS    WHICH    DEPEND    UPON     SEVIILAK    PRINCIPLES.- 
INJECTOE. 


-THE 


The  principle  of  the  ordinary  suction  pump  consists  in  a 
pipe  witli  one  end  in  the  water  and  the  other  connected  with 
a  pump,  by  means  of 
which  a  partial  vacuum 
is  created  in  the  pipe, 
so  that  the  pressure  of 
the  air  upon  the  out- 
side water  forces  water 
up  the  pipe.  For  the 
production  of  the  va- 
cuum different  meth- 
ods have  been  recently 
adopted.  One  of  the 
simplest  and  most  in- 
genious is  that  of  Pro- 
fessor James  Thompson. 
It  consists  in  causing  a 
stream  of  water,  of  con- 
siderable velocity,  to 
carry  away,  in  part,  the  ^^.^  g3 

air  with  it.     "We  give  a 

sketch  of  the  apparatus,  Fig.  83.  From  the  tank  E  a  vertical 
pipe  descends  and  ends  in  a  conical  mouth-piece  at  A,  inclosed 
by  the  spherical  vessel  B.  From  this  we  have  the  diverging 
pipe  F  and  the  suction  pipe  D.  While  now  the  water  flows 
through  A  with  great  velocity,  it  drives  out  the  air  in  F  as  well 
as  in  B.     Air  is  thus  sucked  out  of  the  pipe  CD,  a  partial  va- 

489 


T^ 

j_  _ 

: 

V'^^^ 

-  -pt -= 

=  — — 

— 

ia 


490  THERMODYNAMICS. 

cuum  is  created,  and  tlie  water  rises  in  CD.  Tliis  water  enters 
B  with  a  certain  velocity,  and  is  tliere  carried  out  along  witli 
the  stream  in  F. 

While  here,  we  make  use  of  the  velocity  of  a  stream  of  water 
for  the  formation  of  a  vacuum,  we  might  make  use  of  a  currejit 
of  air  or  steam.  The  latter  is  applied  thus  in  the  blast-pipe 
of  the  locomotive  and  in  Giffard's  injector.  In  the  case  of  the 
blast-pipe,  the  steam  passes  from  the  cylinder  at  about  1.25  at- 
mospheres pressure,  through  a  nozzle  in  the  lower  part  of  the 
stack.  The  velocity  is  by  reason  of  this  pressure  great,  and 
the  steam  drives  out  with  it  the  gases  of  combustion,  and  im- 
parts to  them  a  greater  upward  velocity,  so  that  a  partial  vacu- 
um is  created  in  the  smoke-box  and  the  outer  air  enters  rapidly 
through  the  grate,  thus  causing  more  rapid  combustion.  The 
operation  of  the  blast-pipe  consists  then  in  sucking  in  the  outer 
air  by  means  of  a  current  of  steam.  Giffard's  injector  uses 
steam  for  the  purpose  of  sucking  up  water,  as  in  Thompson's 
water-jet  pump.  It  is  remarkable  that  here  we  can  force  the 
water  thus  sucked  up,  into  the  boiler  from  which  we  obtain  the 
steam.  This  apparatus  is  much  used  instead  of  ordinary  force 
pumps  for  furnishing  feed-water  to  boilers. 

Description  of  the  Injector.— Onv  sketch,  Fig.  84,  shows  a  section 
of  the  apparatus.  The  pipe  A  connects  with  the  steam  space 
of  the  boiler,  and  when  the  cock  H  is  opened  the  steam  passes 
through  a  number  of  holes  in  the  pipe  BG  into  this  pipe, 
which  ends  in  a  conical  mouthpiece  G.  This  mouthpiece  empties 
into  a  chamber  D,  which  communicates  by  the  pipe  FF  with 
the  feed-water  tank  QQ.  The  feed  water  and  the  in  large  part 
condensed  steam,  pass  through  E  and  enter  a  second  conical 
mouthpiece  G.  It  then  passes  through  the  pipe  K,  valve  V 
and  pipe  L,  which  communicates  with  the  water-space  in  the 
boiler.  The  flow  of  steam  is  regulated  by  a  conical  spindle  N, 
which  can  be  raised  or  lowered  by  the  crank  M.  The  flow  of 
feed  water  can  also  be  regulated  by  means  not  shown.  By  the 
pipe  S  the  surplus  water  which  collects  in  the  chamber  R  is 
removed. 

The  action  of  the  apparatus  is  easily  understood.  H  is 
opened  and  the  spindle  G  raised ;  steam  flows  with  great  ve- 
locity into  the  chamber  D,  carries  out  the  air  with  it,  and  thus 


THEORY  OF  THE  INJECTOR. 


491 


causes  in  the  chamber  and  suction  pipe  F  a  partial  vacuum. 
The  feed  water  rises  through  FF  into  the  chamber  D.  The 
spindle  is  then  raised  further,  and  more  steam  enters,  a  great 
part  of  which  is 
condensed  by  con- 
tact with  the  cold 
feed  water.  The 
steam  still  coming 
from  the  boiler 
drives,  by  reason 
of  its  living  force, 
the  feed  water  and 
condensed  steam 
through  E,  into 
the  mouthpiece  G, 
and  it  then  passes 
by^,  F,  audi  into 
the  boiler.  If  the 
ratio  of  the  enter- 
ing steam  and  feed 
water  is  just  right, 
the  stream  of  wa- 
ter where  it  passes 
from  E  to  G,  and 
which  can  be  ob- 
served by  a  small 
window    at    R,  p^^  84 

should  be  perfect- 
'  ly  transparent,  so  that  neither  steam  nor  water  departs  by  S. 

Theory  of  the  Apparatus. — Dimensions. — ^As  has  been  remarked 
in  Part  L,  this  apparatus  testifies  to  the  correctness  of  the 
mechanical  theory  of  heat.  For  if  the  valve  V  is  open,  one 
would  say  that  the  boiler  water  ought  to  escape  just  as  much 
as  the  steam,  since  the  pressure  upon  the  water  is  the  same  as 
the  steam  tension.  If,  however,  we  consider  that  the  work  in- 
herent in  the  steam — in  other  words,  the  total  heat  of  the  steam 
—is  much  greater  than  that  of  the  boiler  water,  we  can  under- 
stand how  the  excess  of  inner  work  can  not  only  suck  up  water 
and  force  it  into  the  boiler,  but  also  heat  this  water  to  the  tern- 


492  TEEBM0DTNAMIC8. 

perature  of  the  boiler.  That  which  in  Thompson's  pump  is 
performed  by  the  outer  living  force  of  the  water,  is  here  per- 
formed by  the  inner  living  force  of  the  steam,  by  that  which  we 
call  the  vibration  work  of  the  molecules. 

Let  the  velocity  with  which  the  steam  passes  the  orifice  at  G 
be  lu ;  the  steam  pressure  in  the  boiler  be  p  kilograms  per 
square  meter  ;  t  be  the  temperature  and  q  the  heat  of  the  liquid. 
Every  kilogram  of  steam  which  passes,  contains  x  kilograms  of 
pure  saturated  steam  and  1  —  re  of  water.  If  G  kilograms  pass 
G  per  second,  we  have  in  this  time  xG  kilograms  of  steam  and 
(1  —x)G  of  water.  We  assume  further,  the  pressure  in  the 
condensing  chamber  Dx  =  ipx- 

According  to  page  474,  the  heat  which  imparts  to  the  steam 
the  velocity  %v  is 

It? 
q  +  xp  +  Apxic  —  {qi  +  cCiPi  +  ApiX^tii)  +  Aff  {p  —  pi)  =  A^^  , 

^9 

where  x  is  the  specific  steam  weight  in  the  boiler,  and  x^  that 
in  the  condensing  chamber.  If  we  assume  that  here  the  steam 
is  completely  condensed,  x^  =  0,  and  we  have 

q  +  xp  +  Apxu  —  qi  +  Aff  (p  —pi)  =  A—.  (LXXIV.) 

Here,  qi  is  the  heat  of  the  liquid  in  the  condensing  chamber, 
hence  q  +  xp  +  Apxu  —  qi,  is  the  heat  which  the  steam  has 
given  up.  We  have  then,  in  this  condensed  steam,  gi  heat 
units  |)er  kilogram.  If  the  flow  is  G  kilograms  per  second,  we 
have 

GA~^Giq  +  xp  +  Apxu  -  q,  +  Aff  (p  -p,)].  (LXXV.) 
From  page  478,  Equation  LXI.,  we  have  also 


'e=^[i-(i^)  "']e. 


tv^  ^  _     mpv 
2g      ~  m 


In  the  following  investigations  and  calculations,  we  shall 
make  use  of  one  or  the  other  of  these  formulae. 


THEORY  OF  THE  INJECTOB.  493 

Let  us  first  apply  tlie  former. 

Tlie  entering  steam  generates  at  first  in  tlie  condensation 
chamber  a  vacuum.  Cold  water  is  then  forced  up  by  tlie  outer 
pressure  of  the  air  into  this  chamber.  If  now  water  flows  from 
a  reservoir,  iipon  whose  surface  there  is  a  pressure  j^q  V^^ 
square  meter,  into  another  in  which  the  pressure  is  p^,,  and  if 
li  is  the  head,  we  have  for  the  living  force  in  the  issuing 
water,  G^, 


;^.=p^+'']^v 


where  u  is  the  velocity  and  -  =  c  ;  hence 

If  h  is  negative,  or  the  water  is  raised,  which  can  only  be 
whenpi  <Poj  we  have 

^  Go  =  [{po  -  Ih)  ff-li\G,.    .    (LXXVI.) 
or 


The  heat  equivalent  to  the  above  work  is 


A^Go=  A  [_{po  -pt)  ff  -  Ji]  Go  heat  units  .     (LXXVII.) 

If  the  heat  of  the  liquid  for  the  cold  water  is  qo,  the  heat  in 
it  is  ^0  Go  heat  units.  At  the  beginning  of  the  entire  process 
we  have  the  heat 

G{q  +  rx  -  ^i)  +  GAff  {p  -p,)  +  A  [{po  -p>i)  ^  ~  A]  Go  +  qoGo- 

(LXXVIII.) 

This  heat  performs  the  following  works  :  1.  It  has  to  heat 
the  cold  feed  water.    2.  It  has  to  impart  to  it  a  certain  velocity, 


494  THERMODYNAMICS. 

SO  tliat  the  water  mass  Gq  and  condensed  steam  mass  G  may 
pass  witli  equal  velocity  through  E.  3.  The  combined  mass 
Gi)  +  G  must  overcome  on  entering  the  chamber  B  the  outer 
air  pressure,  since  B  communicates  with  the  atmosphere  by  S. 
Let  now  the  velocity  of  the  mass  6^  +  6^0  he  Ui,  the  mechani- 
cal work  inherent  in  it  is 

and  this  represents  the  heat 

A'^iG^-  Go)  ....     (LXXIX.) 

Since,  further,  the  Go  kilograms  of  feed  water  must  have  the 
same  heat  of  liquid  as  the  condensed  steam,  it  must  contain 

qiGo  heat  units. 

Finally,  the  work  required  for  overcoming  the  air  pressure 
in  B,  ]po,  is 

{Po-p,)ff{G+G,). 

This  corresponds  to  the  heat 

A{p^  —  l^i)  (5-((9  +  (?i)  heat  units  .     (LXXX.) 

These  three  last  quantities  of  heat  must  equal  that  given  by 
.Equation  LXXVIII.     Hence  we  have 

G{q  +  rx  -  g'l)  +  GA(T{p  -pi)  +  GoA  Kpo  -pi)(^  -  li]  +  qoGo 
=  A  '$  {G  +  Go)  +  q,Go  +  Aff  {po  -p,)  {G  +  Go). 

Or,  reducing  and  canceling  equal  quantities  on  both  sides," 

G{q  +  rx  —  q^  +  GAff  {p  -  po)  -  GqAJi  +  qoGo  = 

a'^  {G  +  Go)  +  q^Go.     .     .     .     (LXXXL) 

This  equation  contains  two  unknown  quantities,  the  heat  q^ 
and  the  velocity  u^.     If  the  apparatus  feeds  the  same  boiler 

from  which  the  steam  is  taken,  the  living  force  A^  {G  +  Go)  of 


THEORY  OF  THE  INJECTOR.  495 

tlie  water  must  be  so  great  as  to  be  able  to  force  the  mass 
{G  +  6^0)  into  the  boiler. 

If  we  assume  that  the  condensation  chamber  D  lies  at  the 
level  of  the  boiler  water,  we  have,  since  the  pressure  in  the 
boiler  is  p  and  thai;  in  R  is  p^, 

"^{G  +  G,)  =  (y {p  - p,){G-  Go). 

The  right  side  represents  the  work  necessary  for  overcom- 
ing the  pressure  p  —  po,  neglecting  all  resistances.     Hence 


Ui  =  V2g0  (p  -po)  =  4.43  Vff  (p  -Po)  .     (LXXXn.) 

where  p  and  po  are  given  in  kilograms  per  square  meter.     If  we 
express  p  and  ^0  in  atmospheres,  then,  since  a  =  0.001, 


th  -  4.43  a/10.334  {p  -  po)  =  14.24  Vp  -  Po- 

By  reason  of  resistances,  we  may  put  Ui  =  14.67  Vp  —  Po,  and, 
since  po  is  always  1, 


n^  =  14.67  Vp  -  1  meters.    .     (LXXXIII.) 

If  in  LXXXI.  we  put  ^l"  ((?  +  G^o)  =  ^  {p-P^)  {G-  +  Oo),  we 
have 

G{q  +  rx  —  q^  +  GAff  (p  —  po)  —  GqAJi  +  qoGo 

=  A(T{p-p,){G  +  Go)  +  q^Go (XC.) 

This  equation  leads  to  interesting  considerations.     "We  may 
consider  it  as  composed  of  two  equations,  viz.  : 

G{q+rx-q,)  =  {q,-qo)Go.     .     .     .     (XCI.) 
and 

GAff  (p  -po)  -  GoAh  =  Ag-  (p  -Po)  {G  +  Go), 
or 
0  =  GoAffip  -  Po)  +  GoAh  =  Go  [ff  (p-p,)  +  h].    .     (XCII.) 


496  THERMODYNAMICS. 

'  The  first  of  these  equations  would  indicate  that  the  heat 
given  up  by  G  kilograms  of  steam  and  water  of  the  temper- 
ature t,  when  cooled  under  constant  pressure  p  into  water 
at  ^1,  is  sufficient  to  heat  Gq  kilograms  from  fo  to  t^  degrees, 
or  to  impart  Gq  {q-^  —  q^  heat  units.  But  if  this  is  so,  then, 
as  is  seen  from  the  second  equation,  there  is  no  heat  or 
mechanical  work  remaining,  in  order  to  raise  the  mass  (tq  and 

G 
force  it  into  the  boiler.      If,  therefore,  the  ratio  -^^  and  the 

(to 

values  of  q,  q^,  and  x  are  known,  we  should  find  q^  from  the 
first  equation,  too  large,  and  the  excess  is  that  heat,  or  work, 
necessary  for  raising  Gq  and  forcing  it  into  the  boiler.  Thus  it 
follows,  unmistakably,  that  in  our  apparatus,  during  the  entire 
process,  a  par,t  of  the  molecular  work  of  the  steam  is  trans- 
formed into  outer  work. 

As  has  been  remarked,  we  can  only  give  a  satisfactory  ac- 
count of  this  apparatus  when  we  assume  this  principle  as 
correct.  That,  however,  a  very  small  amount  of  heat  suffices 
for  the  raising  of  the  water,  can  be  easily  shown  by  a  practical 
example.  This,  also,  follows  from  the  fact  that  one  heat  unit 
corresj)onds  to  a  mechanical  work  of  424  meter-kilograms. 

From  an  experiment  by  the  French  engineer  Villiers,  the 
height  li  to  which  the  injector  raised  the  water  was  4  meters, 
the  pressure  p  in  the  boiler  was  4l  atmospheres  corresponding 
to  a  temperature  ^i  of  146.19'',  and  heat  of  liquid  q  =  147.55 
heat  units,  and  total  latent  heat  of  r  =  503.54  heat  units.  The 
steam  contained  about  7  per  cent,  of  water.  Hence  x  was 
1  -  0.07  =  0.93,  and  rx  =  503.54  x  0.93  =  468.29.  The  tem- 
perature ^0  of  the  feed  water  was  23.5°,  hence  go  =  23.51.  Fi- 
nally, the  temperature  of  the  issuing  water  (mixture  of  steam 
and  water)  was  fi  —  60.5°,  and  hence  q^  =  60.64.  If  we  sub- 
stitute these  values  in  the  first  of  the  above  equations,  we  have 

G 
for  the  ratio  -^  of  the  feed  water  to  the  steam  used, 
Cr 

Go  _  147.55  +  468.29  -  60.64  _  -, .  q^ 
G  ~  60.64  -  23.51 


If,  then,  no  mechanical  work  had  been  necessary  for  raising 
the  water  and  forcing  it  into  the  boiler,  we  should  have  for  the 


THEOBT  OF  THE  INJEGTOB.  497 

given  data  of  tlie  apparatus,  14.95  times  as  mucli  cold  feed 
water  as  steam.  But  now,  for  this  raising  and  forcing,  work  is 
required,  and  hence  this  ratio  shoukl.  be  less.  And  inversely, 
if  this  ratio  were  correct,  the  heat  of  the  liquid  q^  should  be 
less.     The  second  equation  tells  us  by  how  much,  viz.,  by 

A\li  -\-  0  {p  —  ^Jo)]  lieat  units. 

Since  now  h  =  ^,  g  =  0.001,  p  =  10334  x  4i,  p^  =  10334,  we 
have 

jk  [4  +  10334  (4.25  -  1)]  =  0.089  heat  units  per  kilogram. 

Of  this,  4^  X  4  =  0.01  per  kilogTam  are  used  for  raising  the 
water  and  0.079  for  forcing  it  into  the  boiler.  "We  see  that  this 
heat  is  so  insignificant  that  it  can  only  be  observed  by  specially 
constructed  thermometers.  This  is  explained,  as  already  re- 
marked, by  the  fact  that  1  heat  unit  represents  the  considerable 
work  of  424  meter-kilograms,  and  hence  a  very  small  loss  of 
molecular  work  can  cause  a  considerable  amount  of  outer 
work.     For  these  reasons  we  may  use,  in  all  practical  cases, 

C 
the  first  equation  for  determining  ~^ ,  when  i,  fj  and  fo  are  known, 

C 
or  for  determining  t^  from  t,  to  and-^p. 

From  the  equation 

G{q  +  TX  -  <7i)  =  Gq  {qi  -  qo) 
we  have 

G{q  +  Tx)  -F  (7o7o  =  Goqi  +  Gq^,  hence 

_G{q-\-rx)  +  Goqa 
^'~  Go  +  G 

Instead  oi  q  +  rx,  we  can  put  the  total  heat  {i&)  in  1  kilo- 
gram of  steam  of  the  temperature  t  Since,  further,  go  and  q^, 
cannot  be  large  (go  is,  for  example,  on  an  average  15°,  and  q^  at 
most  60°),  we  can  put  the  temperatures  in  place  of  the  liquid 
heats.     We  have  then 


_GW ^  Goto 
^'-     Go  +  G        •    •    •    •     ^^™^-^ 


32 


498  THERMODYNAMICS. 

wliere  W=  606.5  +  0.305^.     We  have  also 
Go  _  W-  h 


G      h-to 


(xciY.; 


Hence  we  have  the  temperature  of  the  mixture  of  steam  and 
liquid.  This  formula  could  have  been  found  directly,  if  we  had 
assumed  the  views  as  to  the  action  of  the  apparatus  referred 
to  as  correct. 

Before  proceeding  to  further  calculations  of  the  dimensions 
of  the  apparatus,  let  us  consider  here  a  question  of  practical 
interest,  viz.,  which  is  the  most  economical,  an  ordinary  feed- 
pump or  the  injector  ? 

We  have  assumed  above  that  the  injector  requires  G  kilo- 
grams of  steam  per  second  of  the  temperature  t.  The  genera- 
tion of  this  steam  out  of  water  at  f,  if  in  each  kilogram  there 
are  x  kilograms  of  steam,  requires  Grx  heat  units.  Now  the 
injector  forces  every  second  G  +  Go  kilograms  of  water  at  ti  de- 
grees into  the  boiler,  in  which  it  is  heated  up  to  t°.  This  re- 
quires the  heat  (Go  a-  G)  {q  —  q^  heat  units.  The  total  heat 
required  per  second  is  then 

Q:=Grx+{Go+G)q-qr) (XCV.) 

Now  we  have  from  the  general  equation,  page  495, 

^0  (^1  -  qo)  +  G^A  [c  {p  -po)  +  ]i]  =  G{q  +  rx  -  qi), 
or 

GoA  [ff  {p  -p^  +  7i]  -  6^0-70  =  Grx  +  G  (q  -  q^)  -  G^q^, 

and,  adding  G^q  to  both  sides, 

G^  {q  -  go)  +  G^A  [(T  {p  -p,)  +  ]{]  =  Grx  +  {G  +  G^)  (q  -  q,). 
From  this  equation  and  Eq.  XCV.,  we  have 

Q  =  Go  {q  -  qo)  +  G,A  [ff  (p  -p,)  +  A].     .     (XCVI.) 

The  first  member  on  the  right  gives  the  heat  which  Gq  kilo- 
grams of  water  require  in  order  to  become  heated  from  fo  to  t, 
that  is,  to  the  boiler  temperature.     The  second  member  is  the 


THEORY  OF  THE  I^^JECTOB. 


499 


heat  required  to  raise  Gq  kilograms  of  T\-ater  to  tlie  lieiglit  h 
and  tlien  to  force  it  into  tlie  boiler.  Hence  the  heat  required  to 
run  the  injector  dejjoicls  only  on  these  quantities,  and  is  independent 
of  tlie  steam  quantity  G  used  hy  the  apparatus.  This  is  evident 
wlien  we  consider  that  the  quantity  of  heat  required  by  the  in- 
jector is  always  returned  to  the  boiler.  If  it  uses  more  steam, 
or  steam  of  higher  tension,  it  can  furnish  more  water,  but  in  all 
cases  the  heat  required  for  the  generation  of  this  greater  ten- 
sion is  returned  to  the  boiler.  Of  course,  in  this,  we  disregard 
all  losses  of  heat  due  to  radiation,  conduction,  and  loss  of 
steam.  If  we  remember  still,  that  the  second  member  on  the 
right  is  almost  vanishingly  small  with  respect  to  the  first,  we 
have 

Q=G,{q-qo)   ' (XCVIII.) 


Hence  the  heat  required  by  the  injector  is  greater  the  more 
water  it  furnishes  to  the  boiler,  or  the  more  water  or  steam  the 
engine  uses  per  second ;  the  greater  the  temperature  of  the 
boiler,  and  the  less  that  of  the  feed  water.  Precisely  the  same 
is  true  of  the  ordinary  feed  pump. 

Sujypose  that  the  pump  has  first  to  raise  the  cold  water  to 
the  height  h.     The  work  required  is  GqIi. 

This  water  is  now  to  be  forced  in-  '^ 

to  the  boiler,  where  the  pressure  is 
p.     This  requires  the  work 

G^pff-  GoPo(}=^  Goff{p—po). 

If,  therefore,  the  pump  makes  one 
revolution  per  second,  it  furnishes 
per  second  Gq  kilograms  of  water  at 
the  temperature  to°,  and  the  work        , 
required  is 

Goh+  Go<j{p  —po), 

or  in  heat  units- 

AGo[ff{p-20o)  +  h'], 


or  the  same  as  in  equation  XCYI.  was  found  for  the  injector. 


500  THERMODYNAMICS. 

In  tlie  boiler,  the  G  kilograms  are  heated  from  U  to  t  degrees, 
which  requires 

6^0  {<!  —  5o)  heat  units. 

The  total  heat  for  the  working  of  an  ordinary  pump  is  therefore 
precisely  the  same  as  for  the  injector. 

Theoretically,  then,  the  one  apparatus  has  no  advantage  over 
the  other.  If  we  consider,  however,  that  the  frictional  resist- 
ances in  the  pump  are  much  greater,  the  injector  is  the  best. 
This  is  not  the  case,  however,  when  the  apparatus  is  used 
simply  for  the  raising  of  water  only,  as  we  shall  soon  point 
out. 

As  to  the  height  of  suction  h,  this,  as  has  been  shown  by  ex- 
periment, is  much  less  when  the  injector  first  begins  to  work 
than  when  in  full  action.  The  reason  may  be  as  follows  : 
When  the  apparatus  is  set  in  action,  we  have  in  the  condensing 
chamber  DD  air  of  atmospheric  tension.  The  steam  rushing 
through  C  carries  with  it  especially  those  particles  near  the 
orifice,  and  causes  a  partial  vacuum.  But  in  consequence  of 
this  there  is  a  quick  vaporization  of  the  particles  of  water  in 
the  steam,  or  of  those  remaining  in  the  condensing  chamber, 
which  diminishes  the  vacuum.  When  once  the  apparatus  is 
in  full  activity,  the  steam  is  at  once  condensed  by  the  cold  feed 
water.  We  have  indeed  still  steam  mixing  with  the  rarefied 
air,  but  steam  whose  pressure  depends  only  upon  the  tempera- 
ture of  the  resulting  mixture  of  steam  and  water. 

According  to  the  exj)eriments  of  Villiers,  at  St.  Etienne,  we 
have  the  following  results  for  the  height  of  suction  and  steam 
pressure  in  boiler  at  beginning  of  action : 

For 

p  =  ±        2.5        3.  3.5        4.        4.5    atmos. 

A  =  1.4      2.  2.47         2.8        3.         3.1     meters. 

When  the  apparatus  was  in  full  action 

A  =3.14    4.  4.49        4.74      4.99    4.99  meters. 

We  see  that  both  in  beginning  as  in  normal  action,  beyond 
a  certain  steam  pressure,  there  is  no  further  increase  in  height 
of  suction. 


THEORY  OF  THE  INJECTOR.  501 

The  results  obtained  by  Beutlier  from  a    steam  jet  pump 
coincide  well  with  the  preceding.     According  to  these  for 


p=^% 

2.5 

3.         atmos. 

h  =  1.32 

1.88 

2.35     meters, 

and  when  in  full  action 

h=  3.45 

3.69 

4.4    meters. 

The  first  results  for  beginning  of  action,  are  given  closely  by 
the  empirical  formula 

7i=  -  1.124  +  lA6x  -  O.W  .  • .     .     (XCIX.) 

where  x  is  given  in  atmospheres. 
For  3  atmospheres  we  have 

h=-  1.124  +  1.46  X  3  -  0.1  X  9  =  2.356  meters. 

These  values  should  not  be  exceeded  for  good  feeding,  other- 
wise the  water  drawn  up  enters  the  condensing  chamber  with 
small  velocity,  and  less  may  be  furnished  than  is  required  for 
feed.  In  general,  Ave  allow  the  water  to  enter  with  about  10  to 
20th  part  of  the  velocity  which  it  possesses  at  U.  If  we  denote 
then  the  velocity  in  the  suction  pipe  by  «,  we  have 

Hence,  the  cross-section  F^  of  the  suction  pipe  should  be 
10  or  20  times  as  great  as  that  of  U.  If  this  is  F^,  we  have 
F2  =  lOFi  to  20i^i.  If  we  denote  the  diameter  by  d^  and  that 
of  E  by  di,  then 

^  =  10^   to  20^, 
4  4  4 


I)'  =  10  to  20. 
If,  therefore,  we  know  the  velocity  21  with  which  the  water 


502  THERMODYNAMICS. 

enters  the  condensing  chamber,  we  can  find  the  pressure  pi 
as  soon  as  we  know  the  height  of  suction  h.  Thus  we  have, 
Equation  LXXYII., 


—  ^j^^a-i^G  —  li     or 


p=i"'-i,-w«  ■■■■■■  '■^■^ 

where  j^i  and  jh  are  given  in  kilograms  ]oer  sq.  meter.      If  2h 
and  po  are  given  in  atmospheres,  and  o"  =  0.001,  we  have 

_  10334  -  lOOOA  -  5121^ 
^'  10334 


^3i  =  1  -  0.097A  -  0.005^2 (CI.) 

If,  for  example,  Ji  =  1.3  meters,  and  w  =  3  meters, 

2h  =  l-  0.097  X  1.3  -  0.005  x  9  =  1  -  0.1711  =  0.83  atmos. 

is  the  pressure  in  the  condensing  chamber. 

As  soon  as  jh  is  known,  we  have  the  velocity  of  efflux  tu  of 
steam 


„  =  443 1/^^  [!_(£)  "']  .  .  (CIL) 
If  we  have  dry  saturated  steam  m  =  1.135,  and 

..  =  1305.83  y^^[l_(^B)"^  J.  .  (Cm.) 
If  the  steam  contains  10  per  cent,  water,  m  —  1.125,  and 


'(/^.[l-(|)'»].    .     (CIY.) 


10  =  1351 


TEEOBY  OF  THE  INJECTOR.  503 

On  account  of  resistances,  we  may  put 


/  m-  \ 

w  =  13245 y  pv\l-  (^\  "'  1 .     .     (CV.) 

m-l 

—  j         is  given  in  Table  IV. 

Since  now  we  can  either  calculate,  or  find  by  experiment,  tlie 
steam  weight  used  per  second  by  a  projected  or  existing  steam 
engine  of  given  liorse  power,  we  can  determine  by  Equation 
XCIY.  the  steam  weight  required  by  the  injector,  if  we  fix 
upon  the  temperature  of  the  mixture  of  cold  water  and  steam 
furnished  by  the  boiler,  and  know  the  temperature  of  the  in- 
jection water.  If  we  denote  the  steam  quantity  in  kilograms 
required  by  the  injector  by  G,  and  if  the  specific  steam  volume 
in  the  boiler  is  v,  we  have  for  the  area  F^  of  the  orifice  C, 
from  Equation  LXY., 


The  diameter  dx  of  this  orifice  is  then 

^i  =  |/^  =  1.129  v:^i. 


EXAMPLE. 

The  steam  pressure  p  in  the  boiler  is  5  atmospheres,  the  height  of  suction 
^  =  1.75  meters.  The  condensing  chamber  is  on  a  level  with  the  boiler-water 
level.  The  engine  uses  7.5  kilograms  of  steam  per  minute  (page  453).  What 
should  be  the  area  of  the  mouthpiece  C  in  a  GifEard  injector  ?  what  of  the  suc- 
tion pipe  F  and  the  pipe  E,  when  the  feed  water  has  a  temperature  of  15°,  and 
the  mixture  of  water  and  condensed  steam  40"  ? 

The  velocity  u-^  of  the  water  in  E  is,  Equation  LXXXIII., 

w,  =  14.67  Vp^^l  =  14.67  4/^=39.34  meters. 

7  5 
Since  the  engine  requires  per  minute  7.5,  or  per  second  -^  =  0.125  kilograms, 


504  THEBMODTNAMICa. 

this  is  the  feed.     Since  the  mixture  of  steam  and  water  has  a  temperature  of  40'', 
we  have  from  Equation  XCIV., 

G^      0.125       w-40 


Q    ~     G    ~  40  -  15  • 

Now  W=  606.5  +  0.305^,  and  if  for  5  atmospheres  is  152.23,  hence  W=  650.33 
and 

0.125  _  650.93  -  40  _  „ ,  ^ 
G     ~  25  -■'^4.44. 

Hence  the  steam  weight  used  per  second  by  the  injector  is 

^  =  ST*^  =  0.00512  kilograms. 
24.44  ° 

We  have  then,  flowing  through  E,  0.125  +  0.00512  =  0.130  kilograms  of 
water  at  40  \  Since  1  kilogram  occupies  the  space  of  0.001  cubic  meter,  0.130 
occupies  0.000130.     If  the  area  of  E  is  F,  then 

Fu,  =  0.000130. 
Since  m,  =  29.34.  _  • 

^  =  0.00000443  sq.  meter. 
If  d  is  the  diameter, 

~  =  0.785cZ2  =  0.00000443,     or    d  =  2.4  miUimeters. 
4 

If  we  take  the  diameter  of  the  suction  pipe  5  times  as  large, 

di  =  12  miUimeters. 

If  dc,  is  5d,  the  water  flows  gV  as  fast  in  the  suction  pipe  as  in  E,   or  has  a 

velocity  of  =  1.174  meters.      Hence  the  pressure  pi,  in  the  condensing 

25 

chamber,  is  from  Equation  CI., 

i?!  =  1  -  0.097  X  1.75  -  0.005(1.174)2, 

or  taking  into  account  resistances  in  the  suction  pipe, 

^j,  =  1  -  0.097  X  1.75  -  0.01  (1.174)2  =  0.82  atmos. 

Now  we  can  find  the  velocity  lo  with  which  the  steam  passes  C.  If  it  con- 
tains 10  per  cent,  water,  we  have  from  CV. 


w  =  lS24..!yi/  „v\  1_  (  ^ 


pv     1 

p 


n 


THEORY  OF  THE  INJECTOR.  505 

Since  v  =  xu  ^  6^  0.90  x 0.363  +  0.001  =  0.328,  and ^  =  -—  —  0.164,  we 

IP  5 

have  from  Table  IV.,  1  -  /^\    '"    =  0.1728,  and  hence 


w 


IV  =  1324.5  y'o  x  0.328  x  0.1728  =  705.96  meters. 

The  cross-section  F^  of.  G  is  now 

„        0.00512x0.328        ^  ,„ 

E,  = ^-^ 0.1821  =  0.0000131,  and 

d^  =  1.129y'5V=  1.129  x  0.00362  z-.  4.09  millinieters. 

Remarli. — ^Altliougli,  as  lias  been  remarked,  the  injector  is  a 
good  feed  apparatus  for  boilers,  and  is  to  be  preferred  to  ordi- 
nary pumps,  tliis  is  by  no  means  the  case  when  the  apparatus 
is  simply  used  for  raising  water.  The  steam  passing  C  has  a 
much  greater  velocity  than  the  water  in  E.  The  condensed 
steam  particles  experience  then  a  sudden  change  of  velocity. 
There  is  thus  impact,  by  which  a  large  part  of  the  living  force 
of  the  steam  is  lost  and  does  not  contribute  to  useful  effect. 
By  this  impact  there  is  indeed  heat,  but  as  in  the  present  case 
M^e  have  to  do  only  with  outer  work,  this  is  of  no  account. 
The  application  of  steam  in  such  a  case  is  no  more  advan- 
tageous than  would  be  its  application  in  the  case  of  an  impact 
or  reaction  wheel. 


CHAPTER  XXIII. 

SUPEEHEATED     STEAM. 

"What  we  understand  by  superlieated  steam  has  been  already 
specified  in  Chapter  XIY.  As  to  saturated  steam,  we  know  that 
it  obeys  different  laws  from  those  which  govern  permanent 
gases.  The  question  arises,  whether  the  same  holds  true  for 
superheated  steam  ?  This  is  in  part  true,  at  least  in  the  vicin- 
ity of  the  point  of  saturation,  that  is,  where  it  passes  into  satu- 
rated steam,  superlieated  steam  differs  in  its  deportment  from 
the  permanent  gases.  Only  when  it  is  far  removed  from  this 
point  are  its  properties  the  same  as  air  or  other  of  the  so-called 
permanent  gases.  In  the  case  of  air,  for  example,  we  have 
learned  that  it  expands  -g-ks  ~  0.00366  of  its  volume  for  each 
degree  rise  of  temperature.  Saturated  steam,  on  the  other 
hand,  when  heated  apart  from  water,  under  constant  pressure 
from  100  to  110°,  expands  5  times  as  much  as  air,  and  for  fur- 
ther heating  from  110  to  115.6,  126.5  and  186.1°,  it  expands  re- 
spectively, 4,  3,  and  2  times  as  much.  It  follows  that  only  at  a 
considerable  distance  from  the  point  of  saturation  is  the  deport- 
ment of  superheated  steam  that  of  a  gas.  "When  it  has  arrived 
at  this  state,  we  have  the  same  uniformity  in  expansion  which 
we  have  for  gases,  and  the  formulse  for  gases  apply. 

This  deportment  of  saturated  steam,  and  of  superheated 
steam  near  the  point  of  saturation,  can  only  be  explained  on 
the  supposition  that  the  molecules  are  more  strongly  attracted 
than  those  of  j)ermanent  gases,  but  that  this  attraction  de- 
creases the  more  the  steam  is  superheated,  or  the  farther  it  is 
from  the  point  of  saturation.  If,  then,  superheated  steam  ex- 
pands under  constant  temperature,  the  heat  imparted  is  not 
equal  to  the  outer  work,  as  is  the  case  with  air,  but  rather 
more  heat  is  necessary  in  order  to  force  the  molecules  apart  or 

506 


SUPERHEATED  STEAM. 


507 


to  decrease  tlieir  mutual  attraction.  If  tliis  lieat  is  not  added, 
tlie  temperature  cannot  remain  constant,  but  tlie  lieat  required 
for  this  inner  work  must  be  furnislied  by  tlie  steam  itself. 


Illustration  hy  Diagram  of  Saturated  and  Superheated  Steam. — 
Let  OA,  Fig.  86,  be  tlie  volume  [S)  of  one  kilogram  of  satu- 
rated steam  at  100^,  and  AB 
tlie  corresponding  pressure 
of  1  atmosphere.  Then  B 
is  a  point  in  a  curve  of  con- 
stant steam  weight.  JlDiD^ 
is  this  curve,  we  have  for 
the  volume 

OA  —  S  =  u  +  a  cub.  m. 


If  OCis  the  volume  of  the 
steam  for  a  less  temperature 
and  CB  the  pressure,  B  is 
also  a  point  in  the  curve, 
and 

OG  —  Si  =  «i  +  0-,  etc. 


I^ARF 


If,  therefore,  from  any  point  B  of  the  curve  B^Bi  we  let  fall 
jEjP,  we  have  the  pressure  for  any  temjDerature.  This  last  can 
be  found  directly  from  Table  II.,  or  calculated  from  the  formula 
of  Magnus  (page  369).  In  like  manner  OF  gives  to  the  scale  of 
abscissas  the  volume  corresponding  to  this  pressure  and  tem- 
perature. If,  now,  we  suppose  a  point  G,  between  the  curve 
B^Bi  and  the  axis,  this  corresponds  to  a  mixture  of  steam  and 
liquid.  The  perpendicular  Gil  gives  the  pressure,  but  Oil  is 
not  the  specific  stemn  volume  (volume  of  one  kilogram),  but  the 
specific  volume  of  the  mixture,  which  we  denote  by 

V  =  XU  +  ff. 

Since  we  can  determine  GH  in  the  same  way  as  BF,  and 
since  for  each  pressure  the  value  of  u  is  known,  and  since  OH 
gives  V,  X  is  given  by 

V  —  ff 


608  THERMODYNAMICS. 

If  we  wish  to  find  tlie  specific  steam  volume  for  the  pressure 
GH,  it  is  only  necessary  to  draw  a  parallel  to  OX  through  G  till 
it  intersects  D^Di,  and  a  perpendicular  from  this  point  cuts  off 
upon  OX  the  required  volume.  All  points,  therefore,  between 
DiD^  and  the  axes  are  points  which  relate  to  a  mixture  of  steam 
and  water.  For  each  of  these  points  there  is  a  curve  of  con- 
stant steam  weight. 

If  we  take  a  point  I  on  the  other  side  of  DiD^,  this  relates  to 
superheated  steam.  The  perpendicular  IK  gives  the  pressure 
and  OK  the  corresponding  steam  volume.  Since  IK  is  greater 
than  LK,  the  pressure  for  the  same  volume  of  saturated  steam, 
the  temperature  must  be  higher.  If  we  draw  IM,  31  is  that 
point  for  which  saturated  steam  of  the  same  pressure  possesses 
the  less  volume  ON.  The  further  I  is  from  the  curve  D^Di,  the 
more  is  the  steam  superheated,  and  the  more  perfectly  the  for- 
mulae for  perfect  gases  apply. 

The  Laiu  of  Him. — We  suppose  in  a  space  v,  1  kilogram  of 
saturated  steam  confined.  Let  it  flow  into  a  vacuum,  or  con- 
ceive the  vessel  enlarged.  Then  the  steam  will  occupy  a 
greater  volume,  and  be  no  longer  saturated.  By  this  operation 
no  outer  work  is  performed.  In  the  case  of  gases,  as  we  know, 
the  temperature  would  be  constant,  since  there  is  no  disgrega- 
tion  work.  Their  inner  work  is  then  unchanged.  Now  Hirn 
asserts  (Zeuner,  Mech.  Warmetheorie,  page  435),  that  for  steams 
also,  from  the  point  of  saturation  up  to  that  luhere  they  have  the  de- 
portment of  gases,  the  EsnsrEE  woek  must  he  constant,  lohen  they  ex- 
pand in  a  vacuum,  or  adiabatically  loithout  performing  older  work. 

There  is  indeed  no  reason,  from  the  standpoint  of  the  mechan- 
ical theory  of  heat,  for  calling  in  question  the  truth  of  this 
principle.  Now,  we  know  that  the  so-called  isodynamic  curve 
is  that  which  gives  the  change  of  condition  when  the  inner 
work  is  constant.  This  curve  then  must  give  the  deportment 
of  steam,  when  it  expands  in  a  vacuum,  from  its  point  of  satu- 
ration up  to  that  where  it  coincides  with  a  gas  in  its  properties. 

Hirn  concludes  further  that  the  law  of  this  curve  is  given  by 

pv  =  piVi  —  P2V2,  etc., 
that  is,  by  the  same  equation  which,  in  the  case  of  gases,  gives 


8UPEBHEATED  STEAM.  609 

tlie  isothermal  curve.  If,  starting  witli  this  principle,  we  cal- 
culate tlie  specific  volume  of  superheated  steam  for  various 
temperatures  and  pressures,  we  find  certainly  a  very  satisfac- 
tory agreement  with  experimental  results.  That  the  principle 
cannot  be  perfectly  valid  will  appear  from  the  following :  We 
have  already  seen  (page  420)  that,  as  shown  by  Zeuner,  the  law 
for  the  isodynamic  curve  of  saturated  steam  is  given  very  ex- 
actly by 

where  n  =  1.0456.  Although  this  value  of  n  is  indeed  not  far 
from  1,  it  follows  that  saturated  steam,  when  it  expands  but  a 
very  little,  or  is  but  little  suj)erheated,  cannot  suddenly  jDass 
into  a  condition  where  the  equation 

pv  =  p^vi  =  ^2^'2 

holds  good.  Near  the  point  of  condensation,  therefore,  the 
isodynamic  curve  of  steam  must  folloAv  a  somewhat  different 
law  from  Hirn's.  It  thus  seems  justified  when  we  assume  that 
in  the  adiabatic  expansion  of  saturated  steam  in  an  empty 
space,  the  law  of  change  of  condition  is  given  by 

where  the  value  of  n  changes  from  n  =  1.0456  to  n  —  1,  which 
corresponds  to  a  perfect  gas. 

Calculation  of  the  Specific  Volume  of  8i(.perheated  Steam  hy 
Hirn's  Laiv. — Let  BF  be  the  curve  given  by 

pv  =  p-^V]_  =  p^v<i, 

or  the  isodynamic  curve  of  superheated  steam  according  to 
Hirn.     We  have  then 

p  '.pr\p2  =V2:v,  :v. 

Through  B  pass  the  curve  GG  oi  constant  steam  weight. 

Suppose  one  unit  weight  of  steam  at  say  5  atmospheres 
tension  {p)  to  expand  in  a  vacuum,  so  that  the  pressure  p-i 
at  the  end  is  only  1  atmosphere.     Then,  according  to  Hirn, 


510 


THERMOD  YNAMIGS. 


the  entire  inner  work  for  tlie  pressure  p^  is  equal  to  that  at  p. 

Denote  this  inner  work  by  Ui.     The  line  Ui  U^,  parallel  to  OX, 

intersects  GG  at  U^.  This 
line,  then,  gives  the  law 
according  to  which  satu- 
rated steam  of  1  kilogram 
and  volume  V4  expands 
tinder  constant  pressure  to 
^2,  and  thus  passes  into 
the  superheated  condi- 
tion. If  the  specific  heat 
is  Cp,  and  if  we  assume  that 
Cp  is  constant  during  the 
expansion  (which,  accord- 
ing to  Begnault's  experi- 
ments, is  nearly  true),  and 
if  the  absolute  temperature 
at  U-i  is  T.2,  and  at  U„  T^, 
we  have  for  the  heat  im- 
parted 

Cp{T,-T,). 

The  change  in  inner  work  caused  by  this  heat,  measured  in 
heat  units,  is 

and  the  outer  work  performed  is 


or  in  heat  units 


Ap,  {V2  -  Vi). 


Now,  according  to  Hirn's  law,  the  inner  work  Ui  in  the  state 
p)-2V2  is  equal  to  that  in  the  state  pv,  or  equal  to  that  at  B.     As 
soon  as  we  know  p,  then  Ui  or  q  +  p  m  known.     We  also  know 
then  the  inner  work  in  the  state  p^Vi,  U^,  or  (/«  +  A- 
We  have  thus 


Hence 


A  (Ml  -u.}  =  q  +  p~{q^  +  p,). 
Cj,{Ti-  T.;)  =  q  +  p-  (q^  +  Pa)  4-  Ajo^  (^'2  -  ^^4). 


SUPERHEATED  STEAM.  511 

pv 

\v  jJ^u^  —  JJU,      ur      t/g  — 

Hence 


But  now  p^v2=pv,     or    v^ 

V2 


=  q  +  P-  iq2+p2)  +  ^  {pv  -p^Vi) 

=  q  +  P  -  {q2  +  P-)  +  ^{pv- PiV^.  ■ 

But  V  =  u  +  ff    and     v^  —  Ui  +  c. 

Hence 

Cp(I\  -  T2)  =  (7  +  p  -  (^2  -  Pa  +  Ap{u  +  0-)  -  ^j?4  K  +  (^) 

=  q-{-  p  -{-  Apu  —  ((^a  +  P2  +  Apiiii)  +  ApG—  Apiff. 

Now  ^  +  P  +  ^^M  is  the  total  lieat  JF  of  1  kilogram  of  steam  in 
tlie  state  pv,  and  gs  +  P2  +  Ap^ti-i  is  the  total  heat  I^Fg  in  the  con- 
dition ^4^4.     We  have  then 

c,{T,-T,)=W-W,  +  A<f{p-p,).    .     (CVI.) 

Instead  of  T^  —  T«,  we  can  put  fi  —  /s.     Further 

W=  606.5  +  0.305^,     and     TV,  =  606.5  +  0.3054,  hence 

W-  W,  =  0.305  {t-t,). 
We  have  therefore 

Cp  (t,  -  t,)  =--  0.305  {t  -  4)  +  Affip  -p.^. 

If,  then,  the  specific  heat  Cp  is  known,  we  have  for  the  super- 
heating of  the  steam  alone  U.2  C^i, 

i^     i^  _  0-305  {t  - 1,)  +  Aff  (p  -p,)  ^    ^    (i^YLl.) 

Cp 

and  for  the  temperature  of  the  superheated  steam  in  the  state 
P%v^ 

t,  =  i,  I  ^'^^^  ^*  ~  ^'^  ^  ^^  ^^  ~^^^    .    (CYIII.) 


512  TEERMOB  TNA  MICS. 

Our  formula  then  gives  us  the  temperature  ^i  of  superheated 
steam,  when  saturated  steam  of  any  pressure  p  and  temperature 
t  expands  in  a  vacuum,  down  to  the  pressure  p2.  We  can  find 
from 

pv  =  p^v^ 

the  volume  v-z  of  the  superheated  steam  of  the  pressure  p)^. 
If  we  subtract  the  volume  i'4  of  saturated  steam  of  the  same 
pressure  ^34  —p2)^&  can  find  by  how  much  the  volume  v^  of  this 
steam  must  be  heated  under  constant  pressure,  apart  from 
water,  in  order  that  it  may  have  the  volume  v^.  An  example 
will  make  this  clearer. 


EXAMPLE. 

We  have  1  kilogram  of  dry  saturated  steam  of  ^^  =  3  atmospheres^  What 
temperature  will  it  have  when  it  expands  in  A'acuo  down  to  1  atmosphere  ?  How 
many  degrees  must  saturated  steam  of  1  atmosphere  be  heated  under  constant 
pressure,  in  order  that  for  the  same  temperature  it  may  have  the  same  volume  ? 
,  For  j9  =  3  we  have  t  =  133.91°  ;  for  jp^  ^1,1^=^  100'.  According  to  Reg- 
nault,  Cp  =  0.4805,  hence 

,        ,.,      0.305(133.91  -  100)  +  ^'^-i  x  O.COl  x  2 
t,  -  100  -  0.4805  "  ' 

or 

t,  =  100  +  21.63  =  121.63^ 


The  temperature  then  falls  from  133.91  to  121.63,  or  12.28°. 
Since  no  outer  work  is  performed  in  the  expansion,  this  loss  of 
vibration  work  must  be  ascribed  entirely  to  the  disgregation 
work. 

The  specific  volume  v  for  ^3  =  3  is 

V  ~u  +  (J  —  0.587, 
hence  we  can  find  v^  from  pv  —  p.iv^,  or 

v^=-  —  == =— -  ^  1.761  cubic  meters. 

The  volume  is  then  three  times  as  great.  The  specific  vol- 
ume of  saturated  steam  of  the  pressure  2h  =JP4  =  1»  is  1.650. 


SUPERHEATED  STEAM.  513 

If,  then,  saturated  steam  at  1  atmospliere  is  heated  under  con- 
stant pressure,  until  its  temperature  is  121.63°,  that  is  until  it 

is  raised  21.63,  its  volume  is  -,'„.,  =  1.067  times  increased. 

l.boO 

If  we  disregard  in  our  formula  the  member  Aff  {p  —  p^^  we 

have  the  very  simple  equation 


If,  for  example,  we  know  from  experiment  what  volume  V2 
one  kilogram  of  saturated  steam  of,  say,  1  atmosphere  tension 
assumes,  when  heated  from  t^  =  100°  to  fi°,  we  can  find  the 
temperature  t  which  1  kilogram  of  saturated  steam  must  pos- 
sess when,  in  exj^anding  in  vacuo  and  cooling  to  t^,  it  has  the 
same  volume.     Thus 

^~ 0305 •     •     •     '     ^^^^-^ 

Thus  Hirn  found  that  saturated  steam  of  1  atmosphere,  when 
heated  under  this  pressure  to  148.5",  occupies  a  space  of  1.87 
{=  V2)  cubic  meters.  How  great  must  be  the  temperature  t  of 
that  saturated  steam  which,  after  expanding  in  vacuo  down  to 
1  atmosphere,  shall  have  the  same  temperature  and  volume  ? 

We  We         t  =  aiS-5~m0.m5  +  W.5  ^  ^^g  ^,_ 
O.oOo 

This  temperature  corresponds  to  a  pressure  of  9.20  atmos- 
pheres. The  specific  volume  of  this  steam  is  0.203  cubic  me- 
ters. Since  at  the  end  of  expansion  the  pressure  must  be  1 
atmosphere,  we  have  from  j^v  =  p^ih 

9.20  X  0.203  =  1  X  vo,     or    v^  =  1.87  cubic  meters, 

or  exactly  as  found  by  Hirn. 

By  means  of  our  formulae  we  can  also  find  what  volume   1 

kilogram  of  saturated  steam,  of  given  pressure,  has  when  it  is 

superheated  to  any  degree  under  constant  pressure.     Suppose 

we  have  1  kilogram  of  saturated  steam  at  3  atmospheres,  whose 

33 


514  THERMODYNAMICS. 

temperature  is  133.9',  and  tliat  we  lieat  it  30^    Wliat  volume 
will  it  then  have  ? 
We  have 

,  _  (163.9  -  133.9)  0.4805  +  133.9  x  0.305  _  -,  q-,  o 
^  -  073^5 ^^^•'^• 

This  temperature  of  181.2''  corresponds  to  a  pressure  of  about 
10.25  atmospheres,  and  a  speciiic  volume  of  0.184  cubic  meters. 
"We  have  then  for  the  volume  v^  required  {pv  =  p^v^, 

10.25  X  0.184  =  3^2,     or    v.  =  0.629. 

If  then  1  kilogram  of  saturated  steam  at  3  atmospheres  is 
heated  30",  it  expands  from  0.586  to  0.629,  or  0.043  cubic 
meter.  It  follows  that  saturated  steam  of  a  high  pressure  ex- 
pands less  for  a  given  superheating  than  that  of  lower  press- 
ure. We  see,  also,  that  we  cannot  use  our  formula  for  great 
degree  of  superheating  and  high  pressures,  because  t  soon  be- 
comes so  great  as  to  exceed  the  limits  of  our  Table  II. 

Let  us  turn  once  more  to  our  figure.  Suppose  1  kilogram 
of  saturated  steam  in  the  condition  VjPsj  ^'^tl  assume  that  it  is 
heated  under  constant  volume  ^3.  If  the  temperature  is  ^3  and 
becomes  by  heating  t^,  we  can  find  the  specific  heat  for  constant 
volume.     Thus  the  heat  imparted  is 

Since  no  outer  work  is  performed,  this  heat  increases  the 
inner  work.  The  inner  work  of  1  kilogram  of  saturated  steam 
in  the  condition  u^p^  is  gg  +  pg  =  Jg,  and  m  the  condition  jh'^Z) 

^2   +  />2  =  ^2. 

Hence 

Cv  {ti  —  is)  —J^  —  Jz- 

Now  the  inner  heat  J2  is  the  inner  heat  at  B,  according  to 
Hirn's  law,  which  we  denote  by  J.     Hence 

or 


SUPERHEATED  STEAM.  515 

We  liave  found  above  for  the  specific  volume  v^  of  1  kilogram 
heated  under  1  atmosphere  21.63^,  v^  —  1.76.  Saturated  steam, 
which  by  expanding  in  vacuo  has  this  temperature,  must  have 
a  pressure  of  3  atmospheres,  or  a  temperature  of  133.91°.  For 
this  temperature  we  have  the  inner  work  J  —  604.47.  This  in- 
ner work  is  possessed  by  the  1  kilogram  of  steam  after  expan- 
sion, when  its  pressure  is  pa  =  1-  We  find  from  Table  11. 
what  temjoerature  saturated  steam  possesses  whose  specific  vol- 
ume is  1.761.  We  find  by  interpolation  about  98.9°.  For  this 
temperature  ^3  we  have  J3—  qs  +  P3=  596.67.     Hence 

_  604.47  -  596.67  _  ^  0.0 
^'~    121.63-98.9     -^•^'^^• 

More  exact  calculation  gives  0.347. 

Just  as  for  gases,  then,  the  specific  heat  for  constant  volume 
is  less  than  for  constant  pressure.     The  ratio  k  is 


Hirn  has  found  for  various  degrees  of  superheating,  the  spe- 
cific volume  of  steam  expanding  under  constant  pressure,  the 
following  experimental  results : 


100°  V,  -- 

=  1.65  (saturated). 

162° 

V2  =  1.93 

118.5 

1.74 

200 

2.08 

141 

1.85 

205 

2.14 

148.5 

1.87 

246 

2.29 

With  these  data  Zeuner  has  computed  by  Equation  CVIII.  a 
table,  in  order  to  see  how  far  calculation  agrees  with  experi- 
ment, and  thus  to  test  the  validity  of  Hirn's  law. 


616  THERMODYNAMICS. 


SPECIFIC   VOLUME   AND   TEMPERATURE   OF   SUPERHEATED   STEAM 
ACCORDING  TO  ZEUNER. 


P  = 


mos. 

V2  =  1.65 

^  =  100 

=  2 

1.72 

113.1 

3 

1.76 

121.6 

4 

1.79 

128.1 

5 

1.82 

133.4 

6 

1.84 

137.8 

7 

1.86 

141.8 

8 

1.87 

145.3 

9 

1.89 

148.5 

10 

1.90 

151.4 

11 

1.91 

154.1 

13 

1.92 

156.7 

13 

1.93 

159.1 

14 

1.94 

161.3 

If  we  compare  the  numbers  in  this  table  with  the  experi- 
mental results  of  Hirn,  we  find  a  good  agreement. 


APPENDIX  TO  CHAPTEK  XXIII. 

THEOEY  OF   SUPEKHEATED   STEAM. 

Zeuner  has  deduced  a  formula  for  superheated  steam*  which  holds  good 
equally  well  for  saturated  steam  also,  which  enables  us  to  find  easily  the  volume 
from  the  pressure  and  temperature,  or  inversely,  and  which  agrees  very  closely 
with  experimental  results.     We  give  here  an  abstract  of  his  article. 

The  formula  to  which  the  discussion  conducts,  is 

A-l 

pv  =  BT-Cp-ir, 
where  B  and  C  and  Tc  are  constants,  whose  values  are 

B  =  '^''     ~     ,       Cf.  =  0.4805,      h  =  1.333,       G  =  193.50, 
Ak 

and  p  is  given  in  kilograms  per  square  meter,  and  v  in  cubic  meters.     We  see 
that  this  equation  differs  from  that  for  permanent  gases 

pv^BT 

k-l  

only  in  the  term  Cp  *    ,       or      C  ^p. 

If  we  use  this  formula  for  saturated  steam,  since  for  a  given  pressure  there  is 
but  one  corresponding  temperature,  we  have  only  to  insert  the  given  p  and  cor- 
responding t,  and  we  can  calculate  v,  the  specific  volume.  The  specific  volumes 
thus  calculated  agree  perfectly  with  those  calculated  from  the  mechanical  theory 
of  heat,  within  ordinary  limits  of  pressure,  from  1  to  14  atmospheres,  as  we  shall 
see  in  the  following  discussion. 

To  use  the  formula  for  superheated  steam  for  a  given  pressure,  we  can  find  v 
for  any  desired  temperature  greater  than  the  corresponding  temperature  for  sat- 
urated steam.  Volumes  thus  calculated  agree  very  closely  with  those  given  by 
Hirn's  experiments,  as  will  be  seen  hereafter. 

The  formula  is  thus  perfectly  reliable,  and  enables  us  to  solve  many  practical 
problems  which  otherwise  are  incapable  at  present  of  solution.     The  importance 

*  Zeitschrift  des  Vereins  dcutscher  Ingenieure,  Bd.  XI.,  p.  1,  1866. 

517 


518  THEBMODYNAMICS. 

of  such  a  formula  cannot  be  overestimated,  and  we  therefore  devote  a  few 
pages  to  a  presentation  of  Zeuner's  method  of  deduction.  We  shall  have  occa- 
sion to  use  the  Calculus,  but  that  need  not  deter  any  reader  from  satisfying 
himself,  at  least  as  to  the  reliability  of  the  formula,  and  familiarizing  himself 
with  its  use. 

It  has  been  often  observed  that  the  use  of  superheated  steam  instead  of  sat- 
urated in  the  steam  engine  possesses  special  advantages.  Observations  and  ex- 
periments, among  which  those  of  Hirn  hold  a  high  place,  point  decisively  to  this 
conclusion,  and  indicate  that  engines  working  with  sui^crhcatcd  steam  are  more 
economical  in  fuel,  and  must  come  more  and  more  into  favor. 

For  more  than  a  decade  experiments  to  determine  the  advantages  of  super- 
heated steam  have  been  made,  especially  in  America  ;  and  when  we  consider  the 
extremely  favorable  results  of  Wethered  with  mixtures  of  saturated  and  super- 
heated steam,  it  seems  remarkable  that  the  iise  of  superheated  steam  has  not  had 
a  more  extended  application. 

Apart  from  certain  practical  difficulties  in  the  application  of  highly  super- 
heated steam,  we  may  find  an  explanation  of  this  in  the  fact,  that  although  all 
experiments  thus  far  have  proved  the  advantages  of  the  use  of  superheated  steam, 
yet  the  degree  of  advantage  is  uncertain.  The  results,  in  this  respect,  are  very 
contradictory,  and  some  are  very  properly  received  not  without  incredulity. 

In  such  a  state  no  amount  of  experiments  can  avail,  because  the  question  can 
only  be  decided  by  a  thorough  knowledge  of  the  physical  properties  of  steam  in 
general.  A  theory  of  superheated  steam  is  also  of  great  scientific  importance. 
We  know  at  present  only  the  deportment  of  such  steam  at  its  two  limiting  states, 
viz.,  at  its  point  of  condensation  when  it  passes  into  the  saturated  condition,  and 
in  the  highly  superheated  state  in  which  its  properties  coincide  with  those  of  a 
permanent  gas. 

The  formula?  of  the  mechanical  heat  theory  for  these  two  conditions  are  en- 
tirely different  both  in  construction  and  method  of  deduction,  and  thus  far  it  has 
not  been  possible  to  deduce  from  the  equations  for  saturated  steam,  or  for  steam 
and  liquid  mixtures,  those  for  permanent  gases  or  reversely,  or  to  represent  the 
deportment  of  steam  m  transition  from  one  to  the  other  of  the  limiting  condi- 
tions. 

Theoretical  investigations  upon  the  deportment  of  superheated  steam  have, 
thus  far,  been  made  by  Hirn  only.  In  what  follows  we  shall  present  the  results 
of  our  own  investigations,  together  with  applications  to  the  most  important 
technical  problems.  We  shall  confine  our  discussion  to  steam  of  water,  but 
there  will  be  no  difficulty  in  extending  the  method  to  other  steams. 

Preliminary  Investigation. — We  denote  by  v  the  specific  volume,  i.  e.,  the 
volume  of  the  unit  of  weight  (one  Idlogram)  of  steam,  by  p  the  specific  pressure 
(pressure  in  kilograms  per  square  meter),  and  by  t  the  temperature  Centigrade. 
If  pressure  and  volume  are  given,  we  can  easily  determine  whether  in  any  given 
case  we  have  to  do  with  pure  saturated,  or  superheated  steam,  or  with  a  mixture 
of  steam  and  liquid.  For  saturated  steam,  whose  volume  we  shall  denote  by  Vj, 
pressure  and  volume  stand  in  a  known  determinate  relation,  and  from  the  prin- 
ciples of  the  mechanical  heat  theory  we  can  calculate  the  volume  v  correspond- 
ing to  the  pressure  p.  If  we  lay  off  for  pure  saturated  steam  (without  admixture 
of  water)  the  volumes  as  abscissas  and  the  corresponding  pressures  as  ordinates,  we 


THEORY  OF  SUPERHEATED  STEAM. 


519 


obtain  a  curve  DD  of  constant  steam  ^Teigllt,  ■which  I  call  "  the  limiting  curve," 
the  coiirse  of  which  we  shall  investigate  hereafter.    For  every  point  of  this  curve, 
or  for  every  pressure  there  is  a  cor- 
responding detei-minate  absolute  tem- 
perature y, ,  known  by  experiment. 

If  now  in  any  given  case  we  lay 
off  the  volume  of  the  unit  of  weight 
as  abscissa,  and  the  pressure  as  ordi- 
nate, and  the  point  thus  found  falls 
on  the  limiting  curve,  we  know  that 
we  have  to  do  with  pure  saturated 
steam.  If,  however,  the  point  a  falls 
in  the  space  between  the  limiting 
curve  and  the  co-ordinate  axes,  we 
have  for  the  same  pressure,  the  same 
temperature  T-^,  but  a  less  volume. 
We  have,   therefore,    a    mixture  of 

steam  and  water.  The  steam  is  of  the  same  character  as  before.  If  x  is  the 
specific  steam  quantity,  ('.  e. ,  the  weight  of  steam  in  the  unit  weight  of  mixture, 
then  1  —  a;  is  the  weight  of  the  water,  and  if  d  is  the  specific  volume  of  the  water 
the  volume  v'  of  the  mixture  is 


v'  =  a;y  1  4-  (1  —  .t)  d, 

and  from  this  we  can  easily  calculate  the  mixture  ratio  x  for  the  given  volume  v'. 
If,  again,  the  point  falls  in  the  space  outside  of  the  limiting  curve,  as  T,  we 
have  to  do  with  superheated  steam.  In  this  case  the  temperature  T>  Ti,  and 
is  not  given  by  the  pressure  p  alone,  but  depends  also  upon  the  volume  v.  This 
relation 

T=F{p,v). 


is  that  which  thus  far  is  unknown  for  superheated  steam,  and  it  is  this  which  we 
shall  first  deduce.  We  shall  call  this  relation  the  equation  of  condition,  or 
"  condition  eqiiation. "  Thus  far  it  has  only  been  assumed  that  the  condition 
equation  takes  the  form 


pv  =  RT 


(1) 


which  holds  good  for  permanent  gases,  in  which  JS  is  a  constant  depending  upon 
the  kind  of  gas,  when  the  point  T,  in  Fig.  88,  lies  very  far  from  the  limiting 
curve,  that  is,  when  the  steam  is  highly  superheated. 

In  the  deduction  of  the  condition  equation  for  superheated  steam,  we  make 
use  of  the  following  principle  of  the  mechanical  heat  theory.  (See  Appendix  to 
Chap.  V.) 

If  the  pressure  p2  and  the  volume  Vo  are  given  for  the  unit  weight,  and  the  body 
expands  or  is  compressed  adiabaticaUy,  the  end  of  the  ordinate  describes  a  curve 
A^A.2,  Fig.  89,  called  the  adiabatic  curve.  If  the  body  is  in  the  condition  a^, 
given  by  the  pressure  js^  and  the  volume  Vj,  the  point  a ^  lies  upon  a  second 
adiabatic  curve  J.  1  J.  I .     If  the  body  passes  from  the  condition  a,  to  the  condi- 


520 


THERMOB  YNAMIC8. 


tion  a,,  and  if  heat  dQ  is  the  heat  imparted  or  abstracted  during  an  infinitely 
small  change  of  condition,  then  /  -^  is  constant,  no  matter  how  the  pressure 

p  changes  with  the  volume  v  during  the  transfer,  that  is,  no  matter  what  the 
curve  a^^i  "^^J  be,  wherever  the  point  a.^  may  be  on  the  first  adiabatic,  or  the 
point  a^  on  the  second.  This  quantity  we  have  called  the  "  heat  weight,"  and  for 
the  sake  of  simplicity  we  denote  it  by  P.  Let  us  now  determine  the  heat  weight 
for  a  mixture  of  water  and  steam. 

In  the  unit  of  weight  of  mixture  let  there  be  x  kilograms  of  steam  at  the 
pressure  p  and  tension  t.  Let  c  be  the  specific  heat  of  water,  which,  according 
to  Regnault,  is 

c  =  1  +  0.00004^  +  0.0000009^3 (2) 

and  let  r  be  the  total  latent  heat,  which,  according  to  Eegnault,  is 


Y    A^    A    Ai 


r  =  60Q.5  +  O.SOot  -  /   cd( .      (3) 

then  we  have,  according  to  Clausius, 

dQ  =  cdt  +  Td 

If  we  divide  both  sides  by  AT,  and 
put,-  for  the  sake  of  simplicity, 


rcdt 
we  have,  using  the  index  2  for  the  condition  a 2  and  1  for  a-^, 

-/II4[(--^)-(-y)]- 


(4) 


(5) 


If  we  assume  that  the  transfer  takes  place  along  the  limiting  curve  DD,  as 
shown  in  Fig.  88,  we  have  for  this  curve 


Xo,  =  1, 


and 


AP: 


and  this  can  be  easily  calculated  for  given  initial  and  final  temperatures. 

For  the  sake  of  simplicity  we  put  ^  =  r  +  -^ (7) 

and  then 

AP^<Py-cp^ (8) 


THEORY  OF  8UPEBHEATED  STEAM. 


521 


Using  Equations  (2),  (3)  and  (4),  we  have  calculated  cp  as  given  by  (7)  for  a 
number  of  values  of  the  pressure,  and  placed  the  results  in  column  3  of  the 
following  Table  I. 

We  pass  now  through  the  two  points 
T.2  and  T^  of  the  limiting  curve 
DD,  Fig.  90,  the  two  adiabatics  A^ 
and  A , .  The  course  of  both  curves 
is  unknown,  since  they  extend  into 
the  space  corresponding  to  super- 
heated steam.  If  now  we  heat  the 
saturated  steam  at  the  temperature 
Tj,  under  constant  pressure  p.,  =  p, 
until  we  reach  the  second  adiabatie  at 
the  point  T,  we  have  for  the  heat 
weight,  when  the  specific  heat  Cp  is 
taken  as  constant, 


log 


Since  the  heat  weight  from  T2  to  Ti  is  the  same,  we  have 
T 


Cp  log 


T, 


Vi 


(9) 


This  formula  holds  good,  to  be  sure,  only  under  the  express  assumption  that 
Cj,,  the  specific  heat  of  steam  for  constant  pressure,  is  constant.  That  this  as- 
sumption is  allowable  is  shown  by  the  experiments  of  Regnault.  He  finds  by 
four  experiments 

Cp  ^-  0.46881,  0.48111,  0.48080,  0.47963, 


and  considers  only  the  first  of  these  values  not  entirely  reliable.  We  have  then 
for  the  mean 

Cp  =  0.4805, 

and  this  value  of  Cp  for  steam  we  shall  assiime  in  all  further  discussions.  We 
shall  see  further  on  that  this  hypothesis  of  the  constancy  of  the  value  of  the  spe- 
cific heat  for  constant  pressure,  and  the  assumption  of  the  correctness  of  Reg- 
nault's  experimental  values,  are  justified. 

By  the  aid  of  Equation  (9)  we  can  find  easily  the  temperature  T  of  super- 
heated steam,  when  the  temperatures  T^  and  T^  are  given,  or  by  taking  different 
values  for  T^  can  calculate  for  a  number  of  points  of  the  adiabatie  through  T^, 
the  temperature  T  of  the  superheated  steam.  Such  a  proceeding  would  lead  to 
little,  and  the  actual  course  of  the  curve  Ai  would  be  in  no  way  known.  We 
must  rather  transform  Equation  (9).  If  we  add  and  subtract  Cp  log  T^  on  the 
left  side,  we  have 


Cp  log 


T, 


<Pi 


9-2 


(10) 


522  TSERMOD  YNA  MIC 8. 

T 
From  this  we  can  find  the  ratio  =-  for  two  points  upon  the  same  adiabatic 

Ai.     If  this  curve  answered  to  a  permanent  gas,  we  should  have  between  press- 
ure and  temperature  of  both  points  the  relation 

T        f 2i\   ^ 


T.-\pJ        ("' 


where  h  is  constant  and  denotes  the  ratio  of  the  specific  heat  of  the  gas  for  con- 
stant pressure  to  that  for  constant  volume. 

I  make  now  the  hypothesis  that  for  superheated  steam,  or  at  least  for  icater- 
steam,  Equation  (11)  also  holds  good,  and  that  here  also  h  is  constant,  but  the 
previous  signification  of  h  no  longer  holds  for  superheated  steam.  The  following 
mathematical  discussion  must  fix  the  more  general  significance  of  the  value  of  k. 

Substituting  Equation  (11)  in  (10),  we  have,  since  in  the  entire  discussion,  as 
shown  in  Fig.  90,  the  pressure  p  is  identical  with  ^2, 

k  -  1  k-\ 

k  m      k 


+    C/,    log     ^^ =       <P2    +    C;,    log    ^^ 


and  hence  it  follows,  in  case  of  the  correctness  of  our  hypothesis,  generally  the 
value 

k-\ 

<p  +  C/.  log  ^^Y~ 


for  saturated  steam  must  be  a  constant.     If  we  denote  this  constant  by  <pn)  we 
have  instead  of  (7) 


T 
(p=Cp  log  -jT^^    +  cpo (13) 


That  this  formula,  with  a  proper  choice  of  the  constants  k  and  <Po.  actually 
gives  the  values  of  (p  with  great  exactness,  may  be  shown  by  the  following  calcu- 
lations. 

We  take  /fc  =  |  =  1.3333,  hence  ^-^^  =  i  =  0.25,   and  cpo  =  1.0933.     Then 
in  common  logarithms 

(P  =  0.2766  logio  —- 1.0933 (13) 

where  |j  is  in  atmospheres,  and  the  temperature  to  be  taken  from  Regnault's 
Tables. 

The  agreement  of  the  values  of  (p,  as  given  by  this  formula  and  by  (7),  is 


THEORY  OF  SUPERHEATED   STEAM. 


523 


shown  in  the  following  table.  The  agreement  might  be  made  still  greater  by- 
taking  k  a  little  different  from  f ,  but  for  the  sake  of  simplicity  in  numerical  cal- 
culations, we  retain  this  round  number. 


TABLE  I. 


1 

2 

3 

4 

Pressure 

Temperature 

* 

in 

Cent,   according 

.  =  -f 

According 

Atmos. 

to  Regnault, 

to 

P- 

Eq.  (13). 

0.1 

46.21 

1.9548 

1.9538 

0.3 

60.45 

1.8929 

1.8912 

0.5 

81.71 

1.8116 

1.8111 

1 

100.00 

1.7519 

1.7520 

2 

120.60 

1.6940 

1.6946 

3 

133.91 

1.6612 

1.6619 

4 

144.00 

1.6384 

1.6391 

5 

152.22 

1.6209 

1.6217 

•     6 

159.22 

1.6071 

1.6076 

7 

165.34 

1.5955 

1.5958 

8 

170.81 

1.5856 

1.5858 

9 

175.77 

1.5769 

1.5769 

10 

180.31 

1.5691 

1.5691 

11 

184.50 

1.5025 

1.5621 

12 

188.41 

1.5563 

1.5557 

13 

192.08 

1.5506 

1.5503 

14 

195.53 

1.5454 

1.5445 

The  great  probability  of  the  correctness  of  our  hypothesis  may  also  be  shown 
in  the  following  manner  : 

If  we  differentiate  Equation  (12),  we  have 


dt  k  —  1  dp 

dcp  =  c^  _-  -  c^  -^  — 


(14) 


If  we  differentiate  Equation  (7),  we  have,  with  reference  to  the  value  of  r  as 
given  by  Equation  (4) 


dcp 


T\dt        T  J 


dt. 


The  expression  in  the  parenthesis  is  the  same  as  that  which  we  have  called 
(Chap.  XVII.,  page  406)  the  specific  heat  of  saturated  steam  for  constant  steam 
quantity,  and  denoted  by  h.  We  can,  therefore,  easily  calculate  it.  We  have 
then 


dcp—  -^dt 


(15) 


524  THEBMODYNAMIGS. 

The  union  of  (14)  and  (15)  gives  then 

h-l  Tdp 


pat (i«) 


We  can  use  this  formula  as  proof  of  the  correctness  of  our  hypothesis. 
Thus  we  have  for  steam  at  the  temperature  0',  100",  200^, 

h=-  1.9166,     -  1.1333,     -  0.6766. 


We  have  from  Regnault's  formula,  which  gives  the  relation  between  pressure 
and  temperature, 

—^  =  19.520,     13.344,     9.851. 
pat 

Inserting  these  values  in  (16),  and  assuming  Cp  as  constant,  and  according  to 
Regnault  0.4805,  we  find  for  h,  for  the  above  temperatures, 

A;  r=  1.3434,     1.3362,     1.8234. 


If,  on  the  other  hand,  we  take,  as  we  have  done,  A;  =  |  =  1.3333,  we  have 
from  Equation  (16), 

<V  =  0-49397,     0.48514,     0.46255, 

and  the  mean  of  these  is  0.4805,  or,  strangely  enough,  exactly  Regnault's  mean 
value. 

From  the  preceding,  then,  we  conclude  that  the  quantities  Cp  and  Tc,  even  if 
future  exact  investigations  may  show  that  one  or  the  other,  or  both,  are  variable, 
vary  very  slowly,  and  may  at  present  be  assumed  as  constant,  with  the  values  as 
given  above.  The  results  of  the  following  investigations  will  further  justify 
these  assumptions. 

Deduction  of  the  Equation  of  Condition  for  Superheated  Steam. — The  "con- 
dition equation  "  must  give  the  relation  between^,  v,  and  t,  or  T=  273-+  t.  If 
we  consider  the  absolute  temperature  as  a  function  of  the  pressure  and  volume. 


-=(f) 

dp  + 

(§)-. 

since  we  can 

,  replace 

)dThjdt, 

"-(1 

'^dp 

^m- 

.  (17) 


THEORY  OF  SUPEBEEATED   STEAM. 


525 


The  fundamental  eqiiations  of  the  mechanical  heat  theory,  as  given  by  Zeu- 
ner,  are 


-=<-->(!)  (I) 

dQ  =  c,dt-AT{^\dp 
If  we  divide  the  last  equation  by  T,  we  have 


dt 


{%h- 


(11.) 


(HI.) 


But  according  to  the  notation  of  Equations  (5)  and  (8),  -^  is  identical  with  d(p, 
and  for  this  latter  we  have  found  according  to  (14) 

di  k  —  1  dp 

Comparing  these  two  expressions,  we  have  for  the  first  new  relation  for  super- 
heated steam 


dt 


k-X 
Inserting  this  in  Equation  L,  we  have,  since  Cp  and  k  are  constant, 

r"  \dp)\  =  k^^' 


(18) 


dv 


and  hence  by  integration 


(dt\  _      Av 


(19) 


wherein  we  assume  indeed  that  the  constant  of  integration,  which  in  general  is  a 
function  ot_p,  is  zero.  This  assumption  will  be  justified  by  the  con-espondence  of 
calculated  results  with  those  of  observation  hereafter. 


526  THERM0DTNAMIG8. 

If  we  make  use  of  (18)  and  (19)  in  Equation  II.,  we  have,  after  easy  transfor- 
mation, 

If  we  determine  Cy  from  this  and  insert  in  (19),  we  have 
fdt\  Av  Tc-1  T 


dpj  ~c^{k-l)'       k      p •    ^^^^ 


while  Equation  (18)  gives 


dt   __      AJcp 
dv  ~~  Cf,  (k  —  1) 


(32) 


The  substitution  of  these  two  in  Equation  (17)  gives  then 


^^  =  ^-TT)  ('^P  +  ^^^^'^  +  -V- 1  ^^  •   •    •    •    (^^) 

and  this  is  the  differential  equation  of  the  equation  of  condition  for  superheated 
steam. 

This  can  be  integrated  easily.     We  have  then 

pv  =  BT-  Cp    ^ (24) 

where  B  and  G  are  constants,  and 

B  =  '^^ (25) 

Now,  for  superheated  steam,  we  have  from  the  preceding  Cp  =  0.4805  and 
k  =  1.333.  Hence  B  =  50.938.  The  other  constant  can  be  easily  determined,  as 
we  shall  soon  see. 

Equation  (24)  is  the  condition  equation  for  superheated  steam.  By  means  of 
it,  when  any  two  of  the  quantities  p,  v,  and  T  are  given,  the  third  can  be  found. 

It  diSers  from  the  equation  for  permanent  gases  only  in  the  term  Cp   k    ,  which 
becomes,  for  superheated  steam,  C  ^Jp  . 

Test  of  the  New  Equation. — If  the  equation  for  superheated  steam  is  correct, 
it  must  hold  for  the  limiting  condition  also,  that  is,  when  the  steam  is  in  the  sat- 
urated condition.  It  should  therefore  give  the  specific  volume  of  saturated  steam 
when  we  substitute  for  a  given  pressure  p  the  corresponding  temperature  t. 
This  leads  us  at  once  to  the  value  of  the  constant  C. 

Thus  the  mechanical  theory  of  heat  gives  for  saturated  steam  of  one  atmos- 
phere tension  {p  =  10334),  and  t  =  100%  or  T—  373°,  the  volume  v  of  one  kilo- 
gram, V  —  1.6508  cubic  meters.  Using  these  values  in  Equation  (24),  we  find 
C=  192.50. 


THEORY  OF  SUPERHEATED   STEAM. 


527 


If  now  the  equation  is  correet,  it  should  give  the  specific  volume  of  saturated 
steam  for  all  other  pressures.     How  far  this  is  so  is  shown  by  Table  II. 

The  second  column  giA'es  the  specific  volumes  for  various  pressures,  according 
to  the  mechanical  theory  of  heat.  The  third  column  is  calculated  from  Equa- 
tion (24).  It  is  to  be  remarked  that  the  values  given  for  B  and  G  hold  good 
when  the  pressure  p  is  given  in  kilograms  per  sq.  meter.  If  we  wish  p  in  atmos- 
pheres, we  have 


pv  =  BT-C  '^p 
^  =  0.0049287,    and    (7  =  0.187815 


The  table  contains  also,  in  the  last  column,  the  values  of  -^  yjy,    of  which 

use  wiU  be  made  in  what  follows. 

We  see  that  the  agreement  is  very  satisfactory,  and  that  we  can  use  the 
above  equation  for  pure  saturated  steam.  Only  for  pressures  of  less  than  one 
atmosphere  is  the  deviation  of  any  account.     For  such  pressures,  however,  the 

TABLE  II. 


Pressure 
in 

Specific  Volnraa  of  Saturated 
Steam 

Values  of 

Atmos. 

byMech. 
Theory  of  Heat. 

by  Eq.  ,24;. 

B    \/W 

0.1 

14.552 

14.677 

21.429 

0.2 

7.543 

7.583 

25.483 

0.5 

3.171 

3.181 

32.043 

1 

1.6504 

1.6506 

38.106 

2 

0.8598 

0.8583 

45.316 

3 

0.5874 

0.5861 

50.151 

4 

0.4484 

0.4474 

53.891 

5 

0.3636 

0.3630 

56.982 

6 

0.3064 

0.3060 

59.640 

7 

0.2652 

0.2650 

61.983 

8 

0.2339 

0.2339 

64.087 

9 

0.2095 

0.2096 

66.002 

10 

0.1897 

0.1900 

67.764 

11 

0.1735. 

0.1739 

69.398 

12 

0.1599 

0.1601 

70.924 

13 

0.1483 

0.1489 

72.357 

14 

0.1383 

0.1383 

73.711 

values  in  the  second  column  are  not  entirely  reliable.  By  the  slightest  change  in 
the  constants  used,  we  might  also  produce  for  such  steam  still  better  agreement. 
It  would  only  be  necessary  to  do  as  Regnault  has  done  in  the  construction  of  his 
formulae  for  the  relation  between  pressure  and  temperature,  and  to  distinguish 
steam  of  more  than  one  atmosphere  pressure  from  that  of  less.  For  our  pur- 
poses it  will  be  sufficient  to  retain  the  constants  as  given  already,  since  we  have 


528 


THERMOD  TNAMIC8. 


in  view  more  especially  the  needs  of  practice,  and  as  in  the  steam  engine  higher 
steam  pressures  are  coming  into  favor. 

From  Equation  (34)  or  (26)  we  now  can  easily  calculate  the  volume  of  super- 
heated steam  for  any  given  pressure  and  temperature.  If  we  take  t  —  100,  110, 120, 
etc.,  and^  =  1  in  Equation  (26),  we  have,  for  example,  lor  superheated  steam  of 
one  atmosphere  tension  the  following  values  for  the  specific  volume  : 


^=100^ 


1.6506  cubic  meters. 


10 

1.6999 

20 

1.7492 

30 

1.7984 

40 

1.8477 

50 

1.8970 

160^ 


1.9463  cubic  meters. 


170 

1.9956 

180 

2.0449 

190 

2.0942 

200 

2.1485 

210 

2.1927 

Hirn  has  observed  the  specific  volume  for  different  pressures  and  temper- 
atures. The  following  table  shows  how  excellently  the  results  of  our  formula 
agree  with  his  experimental  results. 


Pressure 
in 

Temperature. 

Specific  Volume  in 
Cubic  Meters. 

Atmos. 

Hirn. 

Eq.  (26), 

1 

118.5 

1.74 

1.7417 

1 

141 

1.85 

1.8526 

3 

200 

0.697 

0.6947 

4 

165 

0.4822 

0.4733 

4 

200 

0.522 

0.5164 

4 

246 

0.5752 

0.5731 

5 

162.5 

0.3758 

0.3731 

5 

205 

0.414 

0.4150 

If  we  calculate  for  the  same  pressures  and  temperatures  the  specific  volume 
v'  of  air,  the  ratio  of  v  to  v'  gives  the  specific  weight  of  the  steam  with  reference 
to  air.  We  obtain  thus  for  one  atmosphere  and  the  temperatures  100%  150% 
200%  the  specific  weights  0.6401,  0.6316,  0.6250,  that  is,  decreasing  with  increas- 
ing superheating. 

As  another  test  of  the  reliability  of  our  equation,  we  may  determine  the  co- 
eifleient  of  expansion  for  superheated  steam.  If  a  is  the  coefficient  of  expansion, 
we  have  for  gases,  according  to  the  law  of  Mariotte  and  Gay-Lussac, 


pv 


1  +  at 


1  +  ati    ' 
For  constant  pressure^  —  Pi,  and  hence 


rt. 


-  V 


THEORY  OF  SUPERHEATED   STEAM.  529 

For  constant  volume,  Vy  =  v,  and 

Pt\  ~PJ 

The  first  formula  gives  the  coefficient  of  expansion  for  change  of  volume,  the 
second  for  change  of  pressure.  If  we  pass  to  the  difEerentials,  we  have  for 
change  of  volume 

-  =  -W^ •     •     <^^ 

V  -, t 

dv 

and  for  change  of  pressure 


clt        . 


For  a  perfect  gas  both  formulae  give  the  same  value,  a  —  0.003665,  but  not 
so  for  actual  gas  or  steam. 

If  in  Equation  (24)  we  put  for  T,  a  +  t,  where  a  =  273,  and  diflEerentiate,  re- 
garding first  V  and  then  ^  as  constant,  we  have 

dt        pv 

V  —  ^=  - — 
dv       B 

k  —  l 

dt       pv        C  Tc  —  \        jc 
^dp=B^B^—^ 

These  values  substituted  in  Equations  (27)  and  (28),  and  replacing  j)w  by  its  equiv- 
alent in  Equation  (24),  give  us  for  superheated  steam  the  coefllcient  of  expan- 
sion OL  for  change  of  volume 

1 


^-1 


and  for  change  of  pressure 

cc  - ^--iTT^ (30) 


Bh 


These  two  values  are  therefore  different,  and  since  A;  >  1,  the  second  is  always 
somewhat  less  than  the  first.     This  agrees  perfectly  with  Regnault's  observations. 

Also  a  is  always  greater  than  -  =  0.003665,  which  is  also  confirmed  by  experi- 
ment. Further,  ct  is  greater  the  greater  the  pressure  ^,  which  is  also  confirmed 
by  experiment.  Eegnault  has  observed  even  for  hydrogen,  which  approaches 
nearest  to  a  perfect  gas,  for  different  pressures,  somewhat  different  values  for  the 
coefficient  of  expansion. 
34 


530  THERMODYNAMICS. 

From  Equations  (29)  and  (30)  the  following  values  are  computed  for  different 
values  of  ^,  for  superheated  steam.     As^  is  given  in  atmospheres,  the  values  of 


CIO    glYClX    1 

li  ^^j^uauiu-ii  v«"y  o-i-c  uio<:>Li, 

k      -4"- 

Coefficient  of  Expan 

sion. 

Change  of  Volume. 

Change  of  Pressure. 

=  0.1 

a  =  0.003975 

0.003892 

0.5 

0.004150 

0.004017 

1 

0.004257 

0.004090 

5 

0.0C4629 

0.004343 

10 

0.004872 

0.004501 

The  formuliB  above  confirm  all  the  facts  with  reference  to  the  coefficients  of 
expansion  of  gases  and  steams  thus  far  made  known  by  experiment.  We  may 
consider  this,  therefore,  as  a  further  proof  of  the  reliability  of  our  new  formula. 
We  can  deduce,  however,  a  new  result  from  (29)  and  (30),  viz.,  that  the  coefficient 
of  expansion  a  depends  only  upon  the  pressure,  and  not  upon  the  degree  of  super- 
heating or  the  volume.  There  are  no  observations  which  contradict  this  princi- 
ple. If  we  do  not  accept  it  in  its  full  generality,  we  must  at  least  admit  that  it 
is  exact  enough  for  superheated  steam  in  the  neighborhood  of  the  point  of  con- 
densation. 

We  may  now  finally  determine  more  closely  the  specific  heat  of  superheated 
steam  for  constant  volume.     For  perfect  gases  the  value  of  Jc  is  identical  with  the 

ratio  — ^.     This  is  not  so  for  steam.     Here  we  have  Equation  (20) 


Cv  k        Apv 

Making  use  of  Equations  (24)  and  (25)  we  have,  after  reduction, 


f-l+  \^^ (31) 

B  T 

By  means  of  this  formula  we  can  find  for  any  given  condition  of  superheated 
steam  ~- ,  and  then,  since  Cp  is  constant,  the  value  of  c^.  We  see  at  once  from 
the  formula,  that  with  increasing  superheating  the  value  of  —  approaches  k.    For 

Cr>         4 

small  pressure  and  very  great  superheating,  we  should  have  for  steam  —  =  — , 

and  hence  c„  ^  0.3604,  and  this  latter  value  I  regard  as  the  specific  heat  of  steam 
for  constant  volume,  when  by  high  superheating  and  low  pressure  the  steam  has 
passed  into  tho  condition  of  a  permanent  gas.     The  equation  also  shows  that  the 

ratio  —  increases  and  c©  diminishes  the  more  the  steam  approaches  the  saturated 


THEORY  OF  SUPERHEATED   STEAM.  531 

condition.     Thus,  for  example,  Equation  (31)  gives  for  saturated  steam  of  0.1, 
0.5,  1  and  5  atmospheres,  the  following  values  : 


p    =  0.1 

0.5 

1 

5 

^  =  1.358 
c„  =  0.3538 

1.3664 
0.3516 

1.3713 
0.3504 

1.3849 
0.3470 

It  appears  striking  that  the  value  of  the  specific  heat  cv  for  constant  volume 
increases  the  more  the  steam  departs  from  the  saturated  condition.  According 
to  the  usual  conceptions  of  gases  and  steam,  we  shoidd  rather  expect  the  con- 
trary.    Further  investigations  must  confirm  our  result  and  explain  it. 

TJie  Eundamenfal  Equations  of  the  MecJianical  Theory  of  Heat  applied  to 
Superheated  Steam.— If  the  unit  of  weight  of  superheated  steam  is  given  with 
certain  pressure,  volume,  and  temperature,  the  imparting  of  the  heat  dQ  will 
cause  a  change  in  these  quantities.  The  Equations  III.  give  the  relations  gen- 
erally which  subsist  between  these  quantities.  For  superheated  steam  we  have, 
when  we  make  use  of  Equations  (18)  to  (23),  from  these  general  equations 


dQ  =  ^-33  (  vdp  +  Tipdv  j 

/         Tc-1  T      \ 
dQ  =  c„[dt-~^jdp^ 


T     \ 
dQ  =  Co  {dt  +  {k-l)-  dv  1 


These  equations  do  not  differ  from  those  of  permanent  gases  in  form,  but  Cv  is 
here  variable  and  to  be  determined  by  Equation  (20),  while  for  gases  it  is  con- 
stant and  =  -|- . 

If  the  change  of  state  is  reversible,  i.  e.,  if  during  the  change  of  volume  the 
steam  tension  p  is  in  equilibrium  with  the  outer  pressure,  the  work  of  the  expan- 
sion dv  is  pdv,  and  the  corresponding  heat  is  Apdv.  This  portion  of  the  heat  dQ 
goes  then  to  perform  outer  work,  and  the  rest  goes  to  perform  vibration  work 
(rise  of  temperature),  and  disgregation  work,  both  of  which  constitute  the  inner 
work.     If  we  denote  the  change  of  inner  work  by  d  U,  we  have 

AdU  =  dQ  -  Apdv, 

or  replacing  dQ  by  the  first  of  Equations  (32), 


AdU 


A 


k 


l^ipv) „     ,    (33) 


532  THERMODYNAMICS. 

Integrating  this  from  a  given  initial  condition,  we  have 


The  difference  U—Uy  gives  the  increase  of  inner  work,  and  J.  (27—  Z7j)is 
this  difference  measured  in  units  of  heat.  K  we  assume  the  initial  condition  to 
be  water  at  0°,  and  let  J  stand  tov  A{U  —  Z7, ),  we  have 


J=J,  +j^ — -pv (34). 


where  J",,  is  a  constant  to  be  determined,  and  J  shows  how  much  more  heat  is 
contained  in  superheated  or  saturated  steam  of  the  pressure  p  and  volume  v,  than 
in  water  at  0\     We  have  called  JVclq  "steam  heat." 

The  formula  must  hold  good  both  for  superheated  and  saturated  steam.  For 
the  latter  we  know  already  how  to  determine  J,  and  thus  can  find  not  only  the 
value  of  Jq  ,  but  can  test  whether  Equation  (34)  holds  good  for  saturated  steam 
of  any  pressure. 

The  heat  of  the  liquid,  or  the  heat  necessary  to  raise  one  kilogram  of  water, 
under  the  pressure^,  from  0°  to  t"  is 


^Pcdt, 


where  c  is  from  Equation  (2) 

q  =  t+  0.00002^2  +  0.0000003^3. 

If  now  the  water  under  pressure  p  is  completely  changed  into  saturated  steam, 
the  inner  latent  heat  p  must  be  added. 
For  steam  we  have  given 

p  =  575.40  -  0791/. 

Now  for  saturated  steam 

J'=q  +  P (35), 

Thus,  for  example,  for  saturated  steam  of  one  atmosphere,^  =  10334,  t  —  100°, 
t7"=  596.80.  Inserting  this  value  in  Equation  (34)  and  taking  v  =  1.6506,  from 
Table  II.  we  find 

/„=  476.11. 

Using  now  the  values  of  v  in  Table  II.,  we  can  calculate  from  Equation  (34) 
the  values  in  the  last  column  of  the  following  Table  III.,  and  compare  with  those 
obtained  by  Equation  (35). 


THEORY  OF  SUPERHEATED   STEAM. 


533 


TABLE  III. 


Pressure 

Steam  Heat  ./for  Saturated 

in 

Steam. 

Atmos. 

Equation  (35). 

Equation  (34). 

1 

596.80 

596.80 

2 

601.42 

601.62 

3 

604.47 

604.68 

4 

606.81 

606.97 

5 

608.73 

608.81 

6 

610.39 

610.38 

7 

611.86 

611.74 

8 

613.18 

612.96 

9 

614.38 

614.05 

10 

615.49 

615.05 

The  agreement  is  very  satisfactorj^  and  we  see  in  this  a  new  proof  of  the  reli- 
ability  of  our  formula  for  the  deportment  of  superheated  steam. 

We  obtain  very  remarkable  results,  which  afford  further  confirmation  of  the 
correctness  of  our  formula,  when  we  insert  Equation  (24j  in  (34). 

We  have  then 

or  using  the  value  of  B  in  Equation  (25), 


C . 


<hL{T--p 

7c   V         B 


(36). 


For  small  tension  j?  and  high  superheating  we  can  neglect  the  square  of  the 
quantity 

and  thus  obtain  from  Equation  (31) 


Jc  +  ,k  -  1) 


C  p  fc 
B     T 


),  we  have  for  the  same  as- 


sumptions, after  reduction, 

J=J,-\-  (1 


CpJ^_ 
B     T 


CvT 


(37). 


534  THEBM0DYNAMIC8. 

For  very  low  tension  and  very  high  superheating,  in  which  ease  the  steam 
passes  almost  completely  into  the  condition  of  a  permanent  gas,  the  second  mem- 
ber in  the  parenthesis  may  be  neglected,  and  we  have 

J=z  Jo  +  CvT, 

and  this  equation  is  the  same  as  we  actually  have  for  permanent  gases.     If  we 
put  as  above  J=  AU,  where  Z7is  the  inner  work,  then 

AdU=CrdT, 

from  which  we  see  that  the  increase  of  inner  work  is  directly  proportional  to  the 
rise  of  temperature,  a  principle  laid  down  by  Clausius  in  his  Theory  of  Gases. 

Finally,  we  can  determine  for  superheated  steam  that  which  for  saturated 
steam  we  have  called  the  "total  heat"  and  denoted  by  W,  for  which  Regnault 
gives  the  empirical  formula 

Tr=  606.5 +  0.305^. 

This  is  the  heat  which  is  necessary  in  order  to  convert  the  unit  of  weight  of 
water,  under  the  constant  pressure  corresponding  to  the  steam  temperature  t, 
completely  into  saturated  steam. 

If  we  suppose  superheated  steam  of  the  volume  v  generated  under  the  same 
circumstances,  the  work  done  during  its  formation  is  p(v  —  6),  and  the  equiva- 
lent heat  is  Ap(v  —  d),  il  d  is  the  volume  of  the  linit  of  weight  of  water.  We 
can  neglect  d  with  respect  to  v,  and  have  therefore  for  the  total  heat 

W=  J  +  Apv, 
or  from  Equation  (34), 

Ah 

W=  J,  +  j^pv (38). 


Referring  to  Equations  (24)  and  (25) 


k—l 

C 


W^J,  +  c,  [T-'^p     '     j    .     .     .    .    .    (39). 

where  J^  =  476.11. 

Since  this  equation  must  hold  good  also  for  saturated  steam,  we  should  have, 
in  case  this  behaved  like  a  permanent  gas,  C  =  0,  and  then 

W=Jo  +  CpT, 

and  the  comparison  with  Regnault's  formula  would  give  for  the  specific  heat  of 
steam  for  constant  pressure  c^  =  0.305,  which  has  been  obtained  by  Rankine  in  a 
different  manner.  Equation  (39)  shows  clearly  the  reason  of  the  deviation  from 
the  correct  value  Cp  =  0.4805. 


THEOBY  OF  SUPEBEEATEB   STEAM.  535 

Values  of  W  calculated  from  Equation  (39)  for  saturated  steam,  agree  very 
satisfactorily  with  those  found  by  Regnault's  formula.     Thus  we  have  for 

^  =     0.5  1  5    atmos. 

by  Equation  (39),  Tr=  631.15  637.02  653.05 

Eegnault,  Tr=  631.42  637.00  652.93 

Our  Equation  (39)  has  the  advantage  that  it  holds  generally  good  for  super- 
heated steam  also. 

Recapitulation  of  Formula}  Deduced  for  Superheated  Steam. — [Before  pass- 
ing on  to  applications,  we  shall  group  together  here,  for  convenience  of  reference 
and  for  the  benefit  of  those  who  wish  results  presented  in  compact  shape,  the  for- 
mulae deduced  in  the  preceding  for  superheated  steam. 

We  have  for  both  saturated  and  superheated  steam  the  equation 


where 


pv-BT-  Cp  ^ 

B  =  ^iL^^^l) ,     Cp  =  0.4805,      Ti  =  1.333 
Ak  ' 


.     .    (40). 


and  hence  B  =  50.933,  C  —  192.50,  and^  is  in  kilograms  per  sq,  meter. 
If  p  is  in  atmospheres,  we  have 

pv  =  BT-  G  ^p  ) 

[ (41). 

B  =  0.0049287,     G  =  0.187815  ) 

We  have  for  the  ratio  of  Cp  to  Cy 

C  p   k 


1- 


B      T 


Here  we  can  take  p  either  in  atmospheres  or  kilograms  per  sq.  meter  according 
to  the  values  taken  of  G  and  B,  as  given  above.  For  saturated  steam  we  must 
insert,  for  any  given  pressure,  the  corresponding  temperature.  For  superheated 
steam  we  can  insert  any  desired  temperature  greater  than  this.  The  formulae  are 
quite  correct  within  practical  limits  (1  to  14  atmospheres). 

For  the  "  steam  heat "  we  have 


'^='^o+-^4i?'^ 


J^J,^-^^  [T-gp'^' 


(48). 


where  J",,  =  476.11,  and  p  is  to  be  taken  in  hilograms  per  sq.  meters,    k  and  Cr  are 
the  same  as  always,  viz.,  k  =  3-  =  1.333,  Cp  =  0.4805. 


536  TEEBMODYNAMICS. 

For  the  "  total  heat," 

W=J,+c,{t-^P^^ (44). 

were  p  is  in  kilograms  per  sq.  meter,  and  Jq,  Cp,  and  h  as  above. 

The  "  heat  of  the  liquid  "  is  found,  as  always  when  the  temperature  is  known, 

by 

q=^t  +  0mm2t~  +  0.0000003;;' (45). 

or  by  our  steam  tables. 

The  inner  latent  heat  is  found  by 

p-^J-q (46). 

The  outer  latent  heat  by 

Apv=W-J (47). 

By  the  aid  of  these  formulse  we  may  solve  problems  concerning  superheated 
steam  of  great  practical  importance,  and  which  heretofore  have  been  impossible 
of  solution.] 


APPLICATIONS. 


If  we  regard  our  equation  for  the  deportment  of  superheated  steam  as  cor- 
rect (and  from  what  precedes,  the  great  probability  of  its  correctness,  at  least  for 
those  pressures  usual  in  practice,  seems  without  doubt),  we  shall  be  able  to  solve 
many  questions  otherwise  impossible  of  solution.  Especially  easy  of  solution  are 
those  problems  of  practical  importance,  and  a  theory  of  engines  working  with 
superheated  steam  presents  no  longer  any  diiSculty.  We  shall  investigate  here 
a  few  of  the  most  important  eases,  many  of  which,  by  reason  of  known  experi- 
ments in  relation  to  them,  will  serve  as  further  confirmation  of  the  practical  cor- 
rectness of  our  formulae. 

Adiabatic  Curve. — If  the  unit  of  weight  of  superheated  steam  expands  per- 
forming work,  without  heat  being  added  or  abstracted,  the  adiabatic  curve  gives 
the  law  of  variation  of  the  pressure  with  the  volume.  This  curve  gives  also  the 
law  of  the  curve  of  expansion  of  the  indicator  diagram  of  a  steam  engine  working 
with  superheated  steam.  If  the  initial  condition  is  given  bypj,  tu,  T,  (Fig.  91), 
and  the  final  condition  by^jjr  v-i,  and  T^,  we  can  find  by  Equation  (32)  the  rela- 
tion between  these  quantities.  If  in  these  equations  we  make  dQ  =  0,  we  have 
by  Integration 


THEORY  OF  8VPERHEATED  STEAM. 


^2,1 


These  are  the  same  equations  as  for  permanent  gases,  only  there,  for  air  k  = 
1.41,  while  in  the  present  case,  for  superheated  steam,  k  =  1.333. 

The  work  L  during  expansion,  that  is,  the  work  of  each  unit  of  weight  of 
steam,  is 

L  =  I      pdv, 
J  t'j 

or,  replacing p  by  the  relation  pv^  =i'i^'i*  > 


(49). 


where  ^j  is  in  kilograms  per  square  meter. 

The  application  of  this  formula  to  the  steam  engine,  supposes,  indeed,  that 
the  steam  remains  superheated  during 
expansion.  By  great  expansion  it  1^  i9  T. 
may  happen  that  at  a  certain  moment 
the  steam  becomes  saturated,  and 
from  there  on  condenses,  so  that  the 
expansion  curve  is  different.  At  the 
moment  of  the  change,  the  adiabatie 
curve  cuts  the  limiting  curve  DD  at 
the  point  To  (Fig.  91).  The  pressure 
Pa  and  volume  Vq  for  this  point  can 
be  determined.  The  curve  DD  is 
given  by  Equation  XVI.,  Chapter 
XVII.,  viz.  : 

pv"'  =  D, 

where  n  =  1.0646,  and  D  =  1.704,  andp  is  in  atmospheres. 
Since  now  the  point  T„pg  Vq  is  in  both  curves,  we  have 


PoVa 


:p^V, 


Hence 


and    p^.Vo''  =  D. 


(50). 


where  p  is  in  atmospheres. 

This  equation  gives  us  the  expansion  ratio  -— ,  for  which  the  superheated 
steam  just  reaches  by  expansion  the  point  of  saturation.  If  the  actual  expansion 
ratio  -^  is  less  than  this,  the  work  is  given  by  the  simple  Equation  (49).     If  it 

is  greater,  for  instance,  — -,  we  can  find  the  work  up  to  T^,  by  putting  v,,  in 
place  of  t^g  in  Equation  (49).  From  T^  to  2^3  we  must  insert  in  Equation  (49) 
^o'^'g  in  place  of  ^1^1,  and^^  in  place  of  ^-,   and  k  —  1.135  in  place  of  ^  = 


538  THEBMODTNAMTCS. 

1.333,  as  has  been  shown  for  saturated  steam  originally  without  admixture  of 
water  (Chapter  XVIII.,  page  415,  and  Bqu^ion  XXXIII.). 

EXAMPLE. 

An  engine  works  with  superheated  steam  of  j),  =5  atmos.  and  temperature 
1 1  =  180° .  What  is  the  expansion  ratio  when  the  steam  at  the  end  of  expansion 
is  just  in  the  saturated  condition  ? 

The  specific  volume  of  this  steam  is  by  Equation  (26) 

V,  =  0.39037. 
Equation  (50)  gives  then  for  the  required  degree  of  expansion 


—  =  1.332. 


Isodynamic  Curve. — The  isodynamic  curve  gives  the  law  of  variation  of 
pressure  with  volume  where  the  inner  work,  that  is  for  steam,  the  steam  heat,  is 
constant.  If  the  initial  pressure  and  volume  are^,  and  v,,  we  have  from  Equa- 
tion (34) 

A  A 

■p,v,  =  Jo  +  -j^ jpv, 


whence 

p^Vi  =  pv     .     .     . (51). 

"We  see  then,  that  for  superheated  steam,  the  isodynamic  curve  is  an  equi- 
lateral hyperbola,  precisely  as  is  the  case  for  permanent  gases. 
From  Equation  (36),  we  obtain  in  similar  manner 


t,-bP^        =t-sp  ^   ^ 

from  which  we  can  find  the  temperature  Tiov  any  pressure^.  During  expan- 
sion and  fall  of  pressure  there  is  then  an  increase  of  temperature,  while  for  per- 
manent gases  the  temperature  remains  constant.  For  superheated  steam  the 
change  of  temperature  is 

C   /      k~l         k-\\ 
t,-t  =  ^U  —  -p—  ) .     (52). 


This  last  formula  solves  an  interesting  problem.  If  a  vessel  filled  with  steam 
communicates  with  a  vacuum,  the  steam  expands,  and  after  it  has  come  to  rest 
will  be  superheated,  provided  it  was  originally  dry  and  saturated.  The  Equa- 
tions (51)  and  (52)  give  the  final  condition  j3,  v,  t,  for  the  steam  heat  is  evidently 
here  constant.  There  is.  therefore,  a  fall  of  temperature  during  expansion  into 
a  vacuum,  while  for  a  perfect  gas  there  is  no  change  of  temperature. 


THEORY  OF  SUPERHEATED  STEAM. 


EXAMPLE. 

Let  a  vessel  contain  one  unit  of  weight  of  pure  saturated  steam  at  a  pressure 
^,  =  5  atmospheres.  Then  the  volume  is,  by  formula  or  tables,  v^  =0.3630, 
and  temperature  t^  =  152.22°.  Let  the  steam  in  this  vessel  expand  into  another 
in  which  is  a  vacuum,  whose  volume  is  4  times  as  large.  Then  the  final  volume 
is  ^2  =5^1,  and  hence  by  Equation  (51),  ^2  =  1  atmosphere.  From  Equation 
(52),  we  have  for  the  fall  of  temperature  (using  Table  II.), 

t,-to=  18.876° 

Hence  the  final  temperature  t.^  =  133.34°.  As  this  is  greater  than  100°  which 
is  the  temperature  of  saturated  steam  of  1  atmosphere,  the  steam  is  superheated. 

Isothermal  Curve. — If  the  steam  expands  under  constant  temperature,  the  law 
of  change  of  volume  with  pressure  is  given  by  the  isothermal  curve.  For  per- 
fect gases  this  curve  is  identical  with  the  isodynamic,  ^.  e.,  it  is  an  equilateral 
hyperbola.     This  is  not  so  for  steam.     Here  we  have  T  constant  in  our  equation 

pv  =  BT-  Cp~ir^ 

and  hence  the  relation  between  p  and  v  is  given  by  it. 
For  the  initial  condition 


and  by  subtraction 


p^i\  =BT-  Cp  k  ^ 


pV=:piVi  +  Ci^Pt     k      —p    k   ) (53). 


This  is  the  equation  of  the  isothermal  curve  for  superheated  steam,  while  that 
of  the  isodynamic  is  given  by  Equation  (51).  If  both  curves  start  from  the  same 
point,  the  isodynamic  approaches  the  axis  of  abscissas  most  rapidly. 

The  heat  Q,  which  must  be  imparted  to  the  unit  of  weight  of  steam  in  order 
to  keep  the  temperature  constant  during  the  transit  from  ^^  to  ^,  is  found  by  the 
second  of  Equations  (32),  when  we  put  dT=0,  and  integrate.     Thus, 

Q=.c,^TlosP^ (54). 

The  change  of  steam  heat  (inner  work  measured  in  heat  units)  is  from  Equa- 
tion (36) 

'^-  J^^YB\P'  "    -P  "  ) (^^^• 

If  L  is  the  work  of  the  expanding  steam, 


540  THERMODTNAMIGS. 

while  for  gases  Q  =  AL,  and  hence  we  say  that  in  this  latter  case  all  the  heat 
imparted  is  transformed  into  outer  work.  We  see  that  this  is  not  so  for  super- 
heated steam.  Here  only  a  portion  of  the  heat  Q  is  transformed  into  outer  work, 
the  rest  is  required  for  inner  work.     This  portion  is 


c,G 
AhB 


and  the  outer  work  is 


L  =  <ip  -IT  T  log  ;-  -Tm[p^    *    -  i>  * 


Ak    ■"  """^  p    ~  AkB 

Both  quantities  may  be  calculated  from  the  initial  and  end  pressures. 

The  preceding  results  agree  perfectly  with  the  usual  conceptions  as  to  the  de- 
portment of  steam,  but  until  now  it  has  not  been  possible  to  determine  that  por- 
tion of  the  heat  imparted  which  goes  to  perform  inner  work. 


If  saturated  steam  of  5  atmospheres  expands  under  constant  temperature 
down  to  1  atmosphere,  the  heat  imparted  is  found  from  Equation  (54)  by  sub- 
stituting for  -^  =  \,    Cp  z^  0.4805,    and   T  =  273  +  152.23.      We  thus  have 


Q  ==  82.204  heat  units. 

From  Equation  (55),  we  have 

J- J,  =6.803, 

and,  finally,  from  the  difference  of  Q  and  J  —  J^,  the  outer  work  measured  in 
heat  units 

AL  =  75.401, 

or  the  outer  work  itself 

L  =  31970  meter-kilograms. 


Generation  of  Steam  under  Constant  Pressure. — If  one  unit  of  weight  of 
saturated  or  superheated  steam  is  generated  under  constant  pressure  p  from 
water  at  0",  we  have  from  Equation  (89) 


/  G     ^J_\ 


and  the  work  done  is 

pv  =  BT  —  cp    t  . 


THEORY  OF  SVPERHEATED  8TEAM.  54I 

If  the  weight  G  of  steam  is  generated,  we  have  for  the  heat  necessary 

q=g\j^+  c,,  (t--^ irir ^1 (56). 

and  the  work  is 

L=^G{bT-  Cp'k    ). 

If  the  temperature  of  the  steam  for  the  same  pressure  p  is  I\,  and  if  the 
work  (or  the  vohime  of  steam  generated)  is  the  same  as  before,  then  if  G'l  is  the 
weight  of  steam  generated 

G(T--^p-ir)=GAT,-^p-ir\    ,     .     .    (57). 


If,  therefore,  in  any  given  ease  the  pressure  p  and  temperature  T-^  are  given, 
and  we  wish  to  replace  this  steam  by  another  of  temperature  T,  we  can  find  by 
Equation  (57)  the  necessary  steam  weight  G,  and  then,  by  Equation  (56),  the 
heat  requisite  Q. 


EXAMPLE. 

Suppose  we  have  G^  kilograms  of  saturated  steam  of  ^^  =  5  atmospheres,  and 
t\  —  152.22",  and  we  wish  the  same  volume  of  superheated  steam  at  the  same 
pressure  and  the  temperature  f  =  200°.  We  have  from  Equation  (57)  for  the 
weight  of  steam  required  (using  Tables  I.  and  II.), 

6^  =  0.8852  6^,. 

The  heat  required  to  generate  the  saturated  steam  is,  from  Equation  (56), 

Q,  =  653.05  6*, 

For  the  superheated  steam  of  same  volume  and  pressure,  it  is 

Q  ==  676.00  G, 

or,  using  the  relation  between  G  and  G-^ , 

Q  =  0.9163  Q,  . 

The  generation  of  the  superheated  steam,  other  things  being  the  same,  re- 
quires less  heat  than  the  generation  of  the  same  volume  of  saturated  steam,  and 
he7-e  lies  the  advantage  of  engines  ivo?']cing  ivith  superheated  steam.  The  example 
just  given  serves  as  a  direct  comparison  between  engines  of  the  same  size  and 
speed  working  with  saturated  and  superheated  steam  of  5  atmospheres  and  200°, 
provided  there  is  no  expansion. 

There  would  be  no  difficulty  in  extending  the  comparison  to  expansion  en- 
gines also.     "We  shall  only  remark  here,  that  for  such  engines  the  advantage  of 


642  TEEBMOD  TNAMICS. 

superheating  is  somewhat  diminished,  because  the  adiabatie  curve  of  superheated 
steam  approaches  the  axis  of  abscissas  somewhat  more  rapidly  than  that  for 
saturated  steam.  None  the  less,  superheating  is  in  all  circumstances  theoretic- 
ally of  advantage. 

Heating  under  Constant  Volume.  If  the  unit  of  weight  of  steam  is  heated 
under  constant  volume,  only  the  steam  heat  changes,  because  outer  work  is  not 
performed.     We  have,  therefore, 

when  the  initial  condition  is  given  by^, ,  w, ,  5^i,  or  by  Equations  (34)  and  (36), 
Q  =  ^^{p-Pdv, (58). 


■p\    k      )   .     .     .     (59). 


The  first  equation  gives  us  the  relation  between  the  final  pressure  p  and  the 
heat  imparted.     The  second  gives  us  the  final  temperature  T. 

The  preceding  problems  might  be  easily  multiplied.  We  can  easily  solve  for 
superheated  steam  all  the  examples  already  solved  in  this  book  for  saturated 
steam  and  gases. 

Of  especial  interest  are  the  phenomena  of  efflux  of  superheated  steam  through 
orifices  and  the  deportment  of  mixtures  of  steams.  For  lack  of  space  we  only 
call  attention  to  one  more  problem,  which  is  of  importance  because  we  have  in 
respect  to  it  experiments  which  afford  a  new  confirmation  of  the  correctness  of 
our  views  regarding  the  deportment  of  saturated  steam. 

Let  there  be  in  a  cylinder  A  superheated  or  pure  saturated  steam  of  pressure 
Pi,  temperature  T,,  and  volume  Vj.  Let  it  be  forced  under  constant  pressure 
p  I  through  a  pipe  to  a  second  cylinder  B,  where  it  drives  back  a  piston  under 
the  less  but  constant  pressure  p„.  What  is  the  temperature  To,  and  specific  vol- 
ume v.,  when  equilibrium  is  attained  ? 

If  we  follow  the  unit  of  weight  of  steam  from  A  to  B,  we  have  at  the  begin- 
ning the  steam  heat  J^,  and  at  the  end  J.,.  In  A  the  work  isj?|Wi,  and  in  B, 
p.jV^.  The  first  work  causes  an  increase  of  the  steam  heat  Ap^v^.  The  second 
a  diminution  by  Ap^v^.  We  have,  therefore,  if  heat  is  neither  imparted  nor  ab- 
stracted, 

t7"i  +  ApxV'^  —  Ap^v^  =  J-i. 

From  Equation  (34),  we  have 

A  A 

Hence,  after  reduction. 


p,V2  ~  Pl^i 
from  which  we  can  find  the  specific  volume  v^. 


THEORY  OF  SUPERHEATED  STEAM.  643 

If  we  use  this  formula  in  Equation  (24),  we  haA'e 

BT.  -  C>3   k    =  BT,  -  Cih   /-• 
and  hence,  for  the  fall  of  temperature, 

h-h^-'^{p,—  -V,—  \ (61). 


and  from  this  we  can  find  the  temperature  t^. 


If  Pi  =  13  atmospheres,  and  j?2  =  1  atmosphere,  the  fall  of  temperature  is,  by 
Equation  (61)  and  Table  II., 

t,-t.^  =  72.357  -  38.251  =  34.25°. 

If  the  steam  in  A  is  saturated,  then  by  Table  I.,  /^  =  192.08°,  and  hence 
ti  =  157.83°. 

If  the  steam  in  A  is  superheated,  and  has  the  temperature  ^,  =  200°,  205°,  or 
210°,  we  have  for  the  corresponding  temperatures,  since  for  the  same  pressures 
p  I  and  p2  the  fall  is  the  same, 

ti  =  165.75,     170.75,     175.75. 

Hirn  has  found  by  experiment,  for  the  first  ease,  t.2  =  155.58,  and  for  the 
other  three 

ti  =  166°,     171.5°,     177°. 

The  agreement  with  calculation  is  entirely  satisfactory.  For  less  initial 
pressures  p , ,  the  differences  are  greater.  Thus,  when  in  all  cases  the  final  press- 
ure was  Pa  =  1  atmosphere,  we  have  for 

Calculation.  Him. 

^,  =5atmos.     ^,=152.22     ^2=133.34  137.72 

5  246  227.12  238.5 

8  133.91  121.87  128.4 

The  deviations  may  be  ascribed,  for  the  most  part,  to  the  uncertainties  and 
difficulties  of  the  experiments. 

The  preceding  case  occurs  when  steam  escapes  from  a  boiler  into  the  atmos- 
phere. The  temperature  t^  is  then  that  of  the  steam  after  it  has  expanded 
come  to  rest,  and  its  pressure  sunk  to  one  atmosphere.  Of  course  the  experi- 
ment cannot  be  so  tested,  because  the  cold  air  cools  the  steam  jet.  In  order  to 
avoid  this,  Hirn  let  the  steam  escape  into  a  wooden  box  surrounded  by  a  second 
box.  This  second  box  was  inclosed  in  a  third.  The  steam  passed,  after  filling 
the  first,  through  a  large  orifice  into  the  second,  from  this  into  the  third,  and 


544  THEBM0DYNAMIC8. 

finally  into  the  air.  The  apertures  were  so  large  that  the  pressure  in  the  inner 
box,  in  which  the  temperature  t^  was  observed,  hardly  differed  from  the  exterior 
atmospheric  pressure.  It  is  very  desirable  that  these  fine  and  ingenious  experi- 
ments of  Hirn  should  be  repeated  and  extended. 

Since  for  permanent  gases  C  =  0  m  Equation  (61),  the  fall  of  temperature  in 
such  ease  is  zero.  This  can  be  proved  from  the  formulae  m  Chapter  V.,  page 
154.  When,  therefore,  a  permanent  gas  fiows  adiabatically  and  under  constant 
pressure  from  one  vessel  into  another  in  which  the  pressure  is  also  constant, 
there  is  no  change  of  temperature.  This,  of  course,  holds  good  only  for  a  perfect 
gas.  Actual  gases  shows  deviations  similar  to  steam,  as  shown  by  the  experi- 
ments of  Joule  and  Regnault. 

[The  problem  of  the  mixture  of  superheated  steam  with  saturated  steam  is 
one  of  considerable  importance.  It  is  often  the  case  that  only  a  portion  of  the 
steam  passes  through  the  superheater  and  then  mixes  with  the  wet  steam  which 
comes  directly  from  the  boiler  on  its  way  to  the  cylinder.  This  takes  place  under 
constant  pressure,  and  heat  is  neither  imparted  nor  abstracted. 

Suppose  we  mix  together 

O^  kilograms  of  superheated  steam  in  the  condition  p,  i»,,  and  T^,  and 
Gi  kilograms  of  wet  saturated  steam  of  the  pressure^,  specific  volume  w,, 
temperature  To,  and  specific  steam  weight  x, 

Required  the  condition  v  and  T  of  the  mixture,  assuming  that  this  mixture  is 
either  superheated  or  dry  saturated,  i.  e.,  contains  no  water.  (If  we  assume  that 
the  mixture  is  wet,  we  can  solve  the  problem  in  a  precisely  similar  manner,  but 
the  case  does  not  occur  in  practice.) 

We  have  for  the  total  heat  in  the  G^  kilograms  of  superheated  steam,  before 
mixture,  from  Equation  (39), 

and  for  the  total  heat  in  the  G.^  kilograms  of  wet  steam,  before  mixture, 
W,  =  G,  fj-o  +  c,  fT,-^p^^\  -  (1  -  a;)r1 . 
The  total  heat  in  the  mixture  6^  =:  (x^  +  6^3  is,  if  it  is  dry, 

W^g\'Jo  +  c,  {t-  §P^~)1  . 

Since  heat  is  neither  imparted  nor  abstracted,  the  total  heat  after  equilibrium 
must  be  the  same,  hence 

or,  after  reduction, 

GT^zG^Ti  +  G2T2--G^(l-x)^^ (1). 


THEORY  OF  8UPEBHEATEB  STEAM.  545 

From  Equation  (24)  we  also  obtain,  after  reduction, 

Ov^G.v,  +  G^v.^  -  BG.2  {\-x)— (2). 

From  these  two  equations  we  can  find  the  absolute  temperature  Tand  specific 
volume  V  of  the  mixture. 

For  the  change  of  volume  after  mixture,  we  have 

Ov-G^v,  -  a^v.,^- Ba.2{l-x)~      ....     (3), 

pCp 

For  a  given  pressure  then,  the  change  of  volume  is  directly  proportional  to 
the  originally  existing  weight  of  water,  G.,  (1  —  a),  and  it  is  negative,  i.  e.,  there 
is  a  diminution  of  volume  by  mixing. 

In  most  practical  cases  it  is  required  to  find  for  given  values  of  G,  p,  T  and 
T^,  how  much  saturated  steam  Gi  should  be  mixed,  in  order  that  the  resulting 
mixture  may  be  either  superheated  or  dry  saturated. 

We  find  from  (1)  directly,  by  substituting  G^  =  G  —  G^ 

0.=         '^-^'"^     ,. (4). 

T,-T,-{l-i)- 

Op 

If  the  Go  kilograms  were  also  superheated,  we  should  have 

k  —  \' 


W,  =  G,  po  +  cp  (t,  -  ^p  -A^^  J, 


and  hence 

GT=:  G,T,  +  G.T.    ........     (5). 

Gv  =  G^v,    +  G.A'.     ........     (6). 

(r,  -T)G  ' 

^'  -     'T,  -  T,         .•.»..••     (^. 

If  the  G*2  kilograms  were  originally  dry  saturated,  we  should  have  in  (1),  (2), 
and  (3)  a;  =  1,  and  hence  we  should  have  the  same  equations  as  above,  only  in  the 
place  of  lu  we  should  put  u,  the  specific  volume  of  dry  saturated  steam  for  the 
given  pressure. 

In  these  two  eases  theie  is  no  change  of  volume.] 
35 


CHAPTEE  XXIV. 

A. — THE  MORE   IMPORTANT  PRINCIPLES  WHICH   SHOULD   GOVERN   THE 
CONSTRUCTION   OF  THE   STEAM  ENGINE. 

One  of  the  most  important  points  in  the  construction  of  a 
steam  engine  is  that  it  shall  give  a  certain  delivery  with  the 
least  amount  of  fuel.  This  depends  not  only  upon  the  propor- 
tions of  the  engine  itself,  but  also  upon  those  of  the  boiler. 
We  require  from  the  boiler,  first,  that  it  shall  absorb  as  much 
as  possible  of  the  heat  of  the  fuel  and  transmit  it  to  the  water. 
For  this  it  is  necessary  not  only  to  give  the  boiler  an  appro- 
priate shape,  but  also  to  construct  it  of  suitable  material. 
Then  the  furnace  must  be  so  arranged  that  the  fuel  is  com- 
pletely consumed,  and  that  but  little  heat  shall  be  lost  by  radi- 
ation or  conduction.  Sometimes  one  of  these  conditions  is  in 
opposition  to  another. 

As  to  the  form  of  the  boiler,  that  is  to  be  preferred  which 
gives  for  given  capacity  the  greatest  heating  surface.  But  on 
the  other  hand,  this  form  should  give  the  necessary  strength. 
The  first  Watt  boilers,  the  so-called  "  wagon "  boilers,  had  a 
tolerably  large  heating  surface,  and  answered  well  for  the  low 
pressures  then  in  use.  At  present,  when  higher  pressures  are 
used,  they  would  not  be  sufficiently  strong.  Hence  cylindrical 
boilers  are  now  used,  either  with  interior  or  exterior  fire-place. 

The  boiler  should  also  have  such  capacity  as  to  furnish  the 
steam  required  by  the  engine,  and  to  keep  the  engine  in  uni- 
form action.  For  this  reason  the  steam  used  per  stroke  should 
be  but  a  small  part  of  the  boiler  capacity.  In  general,  the 
steam  space  should  be  at  the  very  least  12  times  the  capacity 
of  the  cylinder.  In  order  that  the  heating  surface  may  be 
large,  the  water  should  occujDy  a  certain  extent  of  the  boiler. 

546 


STEAM  ENGINE— GENERAL  PBINCIPLE8.  547 

111  general,  tlie  water  space  is  ^V  of  the  entire  capacity.  In 
order  to  prevent  radiation,  the  boiler  may  be  covered,  where 
exposed,  by  poor  conductors.  Boilers  are  sometimes  con- 
structed now  of  steel  plate  as  well  as  iron,  because  the  former 
is  not  only  stronger  but  has  a  greater  conducting  power. 

As  to  the  furnace,  care  must  be  taken  to  secure  complete 
combustion  of  fuel,  that  the  heat  may  be  absorbed  by  the 
boiler  sides,  and  that  but  little  heat  is  lost.  For  complete 
combustion  a  certain  amount  of  air  is  essential.  But  if  more 
air  than  necessary  is  used,  the  excess  absorbs  a  portion  of  the 
heat,  of  which  indeed  a  part  is  given  up  to  the  boiler,  but 
another  part  escapes  at  the  chimney.  A  good  draught  is  also 
necessary.  This  will  be  greater  the  higher  the  chimney  and 
the  greater  the  difference  of  temperature  of  the  air  in  the 
chimney  and  the  cold  air  outside.  The  height  of  chimney  has 
a  limit,  both  by  reason  of  cost,  and  because  the  increased  fric- 
tion diminishes  the  draught.  The  temperature  in  the  chimney 
should  not  be  too  great,  because  then  a  great  part  of  the  heat 
passes  off  unutilized.  It  has  been  sought  to  utilize  this  waste 
heat  in  the  chimney  by  making  it  heat  the  feed  VN^ater,  when 
ordinary  feed  pumps  are  used.  Engines  working  with  super- 
heated steam,  of  which  there  are  but  few,  use  this  heat  to 
superheat  the  steam. 

In  stationary  engines  the  grate  surface  is  a  certain  propor- 
tion of  the  heating  surface,  about  ^V  or  -^  only.  In  locomo- 
tives this  ratio  is  still  less,  even  as  low  as  -^j^  or  less,  but 
here  there  is  a  strong  artificial  draught.  It  is  thus  pos- 
sible with  a  boiler  of  relatively  small  capacity  and  weight 
(weight  of  boiler  with  water)  to  generate  in  a  short  time  a  con- 
siderable amount  of  steam,  a  property  which  is  of  importance 
in  locomotives  especially. 

The  ratio  of  the  heat  absorbed  by  the  boiler  in  a  given  time, 
as  one  hour,  and  which  can  be  determined  evidently  by  the 
amount  of  water  vaporized  in  that  time,  to  that  furnished  by 
the  fuel,  is  called  the  efficiency  of  the  boiler.  The  heat  units 
furnished  by  the  complete  combustion  of  different  fuels  have 
been  determined  by  experiment.  We  may  call  this  the  heat- 
ing value  of  the  fuel. 

The  more  water  is  evaporated  in  a  given  time  by  a  given 
weight  of  fuel  the  greater  the  efficiency. 


548  THERMODYNAMICS. 

Example  1. — In  a  hot-air  engine  the  heat  furnished  per  hour  co  the  air  is 
6170  heat  units,  wliile  in  the  same  time  4.585  kilograms  of  coal  are  consumed, 
whose  heating  value  is  3500  heat  units.     What  is  the  furnace  efficiency  ? 

We  have 

4.585  X  3500  ~    '^' 
Therefore  62  per  cent,  of  the  heat  is  lost. 

Example  2. — The  boiler  of  an  expansion  engine  which  uses  steam  of  5  atmos- 
pheres, vaporizes  per  hour,  for  every  horse  power,  30  kilograms  of  water,  and  re- 
quires for  this  5  kilograms  of  hard  coal,  whose  heating  power  is  7500  heat  units. 
What  is  the  boiler  efficiency  ? 

If  we  assume  640  heat  units  to  1  kilogram  of  steam  at  all  pressures,  v^'e  have 
30  X  640  =  19200  heat  units  imparted  to  the  water  per  hour  per  horse  power. 
The  5  kilograms  of  coal  give  7500  x  5  =  37500  heat  units.    Hence  the  efficiency  is 

As  to  tlie  engine  itself,  the  first  requirement  is  tliat  for  a 
given  power  it  shall  use  as  little  steam  as  jDossible.  This  is 
accomplished  principally  by  using  the  steam  expansively  and 
having  as  much  expansion  as  possible.  Since  the  counter- 
pressure  upon  the  piston  has  considerable  influence,  this 
shoiild  be  as  small  as  may  be.  Where,  then,  water  is  plentiful, 
condensing  engines  are  of  value.  In  order  that  the  useful 
effect  for  a  given  steam  consumption  may  be  a  maximum,  the 
prejudicial  resistances,  friction,  work  of  the  pumps,  etc.,  should 
be  a  minimum.  These  conditions  require  the  construction  to 
be  simple.  If  we  use  high  steam  (7  or  8  atmospheres)  and  a 
high  expansion  (1  to  6  or  1  to  8)  the  use  of  the  condenser  offers 
less  advantage,  as  the  influence  of  the  back  pressure  is  rela- 
tively less,  and  two  pumps  must  be  worked  by  the  engine. 

While  seeking  to  reduce  the  cost  of  working  to  a  minimum, 
we  should  also  make  the  cost  of  construction  small.  This,  as 
well  as  the  cost  of  erection,  depends  upon  the  dimensions, 
which  we  must  therefore  make  as  small  as  possible.  This  may 
be  effected  by  the  use  of  high  steam,  and  also  by  rapid  action. 
As  both  these  increase,  the  cylinder  volume  becomes  less.  For 
a  rapid  engine,  the  fly-wheel  also  is  lighter  and  the  friction  of 
the  shaft  is  less.  Most  industrial  purposes  also  require  a 
high  velocity,  so  that  high  piston  speed  causes  simpler  gear- 
ing. 


STEAM  ENGINE-GENERAL  PRINCIPLES.  549 

Total  Delivery — Useful  Effect — Efficiency. — By  total  delivery 
we  mean  the  work  of  the  effective  steam  pressure,  or  pressure 
of  boiler  steam  minus  the  back  pressure  of  the  air  or  conden- 
ser. From  this  total  effect  we  have  to  subtract  the  losses  due 
to  difference  between  boiler  and  cylinder  pressure,  friction  of 
piston,  valves,  etc.  The  difference  is  the  calculated  or  theo- 
retical useful  effect.  This  then  is  the  work  actually  imparted 
to  the  engine.  If  we  measure  the  work  done,  by  the  dyna- 
mometer, we  have  the  actual  or  observed  useful  effect.  The 
more  reliable  the  coefficients  used  in  determining  the  losses, 
the  better  the  agreement  between  the  calculated  and  the  ob- 
served useful  effect.  The  division  of  the  useful  effect  by  the 
total  gives  the  efficiency.  A  machine  is  more  nearly  perfect 
the  nearer  this  ratio  is  to  unity.  The  same  method  of  calcula- 
tion is  used  when  we  make  use  of  the  principles  of  the  me- 
chanical theory  of  heat,  as  when  we  proceed  according  to  the 
old  method  of  Pambour,  only  we  have  to  take  into  account 
a  new  loss,  which  Zeuner  calls  the  loss  by  reason  of  the  incom- 
Ijleteness  of  the  cycle  process.  We  shall  return  to  this  later 
on. 

In  any  steam  engine,  the  greater  the  efficiency  of  the  furnace 
and  the  engine  itself,  the  better  is  the  machine.  If  in  addition, 
cost  of  erection  and  repairs  is  small,  all  conditions  are  satisfied 
which  can  be  demanded  of  the  construction.  To  demand  that 
the  furnace  shall  absorb  all,  or  the  greatest  part  of  the  heat 
contained  in  the  fuel,  is  as  unreasonable  as  to  demand  that  a 
water-wheel  shall  receive  the  entire  flow  of  a  river  from  the 
source  to  the  sea. 

If  we  divide  the  useful  effect  of  a  steam  engine,  expressed  in 
units  of  heat,  by  the  number  of  heat  units  given  by  the  com- 
bustion of  the  fuel,  we  obtain  the  "  thermal  effect "  of  the  entire 
apparatus. 

Example  1.— The  total  delivery  of  a  steam  engine  is  1000  meter-kilograms  per 
second,  and  the  useful  effect  537.     What  is  the  eflfieiency  ? 

We  have  ^  =  ^-'^'' 

Example  2. — What  is  thermal  effect  of  the  hot-air  engine,  page  548,  when 
the  hourly  delivery  is  265680  ? 


550  THERM0DTNAMIC8. 

Since  in  1  hour  4.585  x  3500  heat  units  are  set  free  in  the  furnace,  and  265680 
meter-kilograms  corresponds  to  "^^-j^r-  =  626.6  heat  units,  we  have 

,  P-QK — ^7i?r  =  0-039,  or  about  4  per  cent. 
4.585  X  3500  '■ 

The  thermal  effect  is  that  which  properly  informs  us  as  to  the  economy  of 
steam  or  hot-air  engines. 


B. — The  Cycle   Peocess  of   the  Peefect  Steam  Engine,  and 
THE  "Disposable  Woek." 

In  tlie  first  part,  we  have  seen  tliat  tlie  delivery  of  every 
caloric  engine  is  given  by 

wliere  -^^  is  the  heat  weight  imparted  and  T^  —  T  the  temper- 

ature  fall,  or  the  difference  of  the  highest  and  lowest  tempera- 
tures of  the  air  when  compressed  adiabatically.  We  called  it 
there  the  "  useful  delivery,"  because  it  was  that  obtained  by 
subtracting  from  the  total  delivery,  or  work  of  the  air  on  the 
piston,  that  required  for  the  compression  of  the  air  by  the  feed 
piston.  We  shall  now  call  it  the  "  disposable  luorl',"  since  it  is 
that  which  the  air  in  passing  through  its  cycle  puts  at  our  dis- 
position, from  which  we  are  to  get  as  much  useful  effect  as  we 
can. 

We  have  also  seen  what  the  cycle  process  is  when  the  ahso- 
lufe  maximum  of  work  is  required.  The  addition  and  abstrac- 
tion of  heat  must  be  so  regulated  that  all  the  heat  imparted 
must  be  transformed  into  work.  In  other  words,  heat  addition 
and  abstraction  must  take  place  according  to  the  isothermal 
curve,  and  the  two  others  must  be  adiabatic.  Such  a  cycle 
process  we  can  call  "  perfect,"  and  an  engine  which  goes 
through  such  a  cycle  is  called  ''perfect,''  or  ideal.  It  is  impos- 
sible, with  the  same  expenditure  heat  and  temperature  fall,  to 
obtain  a  greater  delivery  than  such  an  engine  gives.  But  it 
has  the  disadvantage  that,  other  things  being  the  same,  it  re- 


STEAM  ENQINE.— GENERAL   PEINCIPLE8. 


651 


quires  a  miicli  greater  cylinder  volume  than  liot-air  engines  in 
wliich  lieat  addition  and  abstraction  take  place  according  to 
some  otlier  law  than  the  isothermal.  For  this  reason  it  is  not 
to  be  recommended  in  23ractice. 

The  case  is  different  in  this  respect  when  we  use  steam  in- 
stead of  air  or  a  permanent  gas.  Here  also  such  an  engine  is 
perfect  when  the  cycle  process  is  perfect.  But  here  such  a 
process  is  the  easiest  executed,  because  the  isothermal  lines 
are  parallel  to  the  axis  of  X 

We  shall  first  speak  of  the  perfect  cycle  process  of  the  steam 
engine.  We  shall  see,  as  we  proceed,  why  in  our  present  en- 
gines the  cycle  is  incomplete.  The  work  of  such  a  j)erfect 
steam  engine  we  call,  with  Zeuner,  the  disposable  work. 

Let  EF,  Fig.  92,  be  the  steam  cylinder  with  the  piston  KK. 
Left  of  the  piston  is  a  certain  weight  of  water  of  G  kilograms. 
The  pressure  upon  the 
piston  is  p,  and  the  back 
pressure  is  j^i.  In  a  con- 
densing engine  2h  is  about 
0.15,  and  in  a  non-condens- 
ing 1.1  atmosphere.  Now 
let  heat  be  imparted  till 
the  water  is  raised  to  its 
boiling  point  t°  for  the 
pressure  p.  If  heat  is  still 
further  imparted,  steam  is 
generated  of  the  pressure 
P  and  temperature  t,  and 
the  piston  is  driven  toward 
the  right.  When  the  dis- 
tance  HV  =  AB  is  passed 

let  the  greatest  part  of  the  water  be  vaporized.  Let  the  specific 
steam  weight  be  now  x.  Then  we  have  now  in  the  cylinder 
Gx  kilograms  of  steam,  and  G  (1  —  x)  of  water.  The  heat  im- 
parted to  the  water  at  f,  to  generate  the  steam,  is  Grx  heat 
units,  where  r  is  the  total  latent  heat  of  vaporization.  Let 
now  the  mixture  expand  adiabatically  along  BC,  until  the  vol- 
ume is  Vi,  and  the  temperature  and  pressure  t^  and  p^.  As  we 
know,  steam  condenses  during  the  expansion,  and  the  specific 
steam  quantity  at  the  end  is  less  than  at  the  beginning.     Sup- 


552  THEBM0DYNAMIC8. 

pose  this  quantity  is  a^i,  tlien  we  have  (page  175^  Equation 

XXV., 

xr  Xii\ 

^  +  ^  =  -^  +  n, 

where  r-^  is  the  latent  heat  of  vaporization  at  the  temperature 
i^i.  The  piston  is  now  at  the  end  of  its  stroke.  Now  let  the 
volume  Vi,  of  the  temperature  t^^,  be  compressed  under  the  con- 
stant pressure  2h  till  the  volume  is  v^,  that  is,  the  piston  passes 
through  GD.  Then  let  the  remaining  volume  be  compressed 
adiabatically,  D  being  so  chosen  that  during  compression  from 
V  to  H,  the  remaining  steam  is  converted  into  water,  and  we 
have  the  original  condition  again.  If  the  process  is  thus  per- 
formed as  indicated,  we  have  not  only  a  complete  cycle  process, 
but  also  Si  perfect  cycle  process,  that  is,  one  in  which  the  work 
obtained  is  a  maximum.  Whatever  other  complete  cycle 
process  the  steam  may  be  made  to  perform,  the  work  obtained 
for  the  same  amount  of  heat  imparted  will  be  less.  If  the  spe- 
cific steam  quantity  at  D  is  x^  (of  course,  less  than  at  (7),  we 
have,  since  at  A,  x^  =  0, 

-jT  +  T^i—  T. 

At  0  the  steam  weight  was  Gx^,  and  at  D  it  is  Gx2,  so  that 
from  G  to  I)  the  heat  abstracted  is 

Qi  =  On  {oc,  -  x^). 

The  entire  process  thus  is  similar  to  that  on  page  232  of 
Part  I.  where  air  expanded  and  was  compressed  under  con- 
stant pressure.  Just  as  there  the  work  actually  obtained,  after 
subtracting  that  of  the  back  pressure  ^5,  is  given  by  the  area 
TiT^TsT^,  so  here  the  area  ABCD  is  the  effective  work, or  that 
obtained  after  subtracting  that  of  the  back  pressure  p^.  If  we 
denote  the  work  by  L,  we  have 

L=:\{Q-Q,)  (see  page  197,  Part  I.) 
A 

If  for  Q  and  Qi  we  put  the  values  above, 
L  =  Grx  —  G7\  {xx  —  X2). 


STEAM  ENGINE— GENERAL  PRINCIPLES.  553 

Since 


na?i  =  i^+r-  T,)  T„     and    nx^^  =  (r  -  r,)  T, 


we  liave 


L  =  ^{T-T,).    .    .    .     (CXIII.) 


L=-^^iT-T,).    .    .     .     (CXIV.) 

This  is  the  same  equation  wliicli  we  found  in  Part  I.  for  the 

delivery  of  the  hot-air  engine.     The  quotient  ^™  is  the  heat 

A  J. 

weight  and  T  —  T^  the  temperature  fall.  Just  as  in  hydraulics 
we  determine  the  total  delivery,  or,  as  we  now  call  it,  the  dis- 
posable work,  of  a  water-wheel  from  the  weight  of  water  enter- 
ing the  wheel  in  a  certain  time  and  the  fall,  so  here  the  dis- 
posable work  of  a  steam  engine  is  given  by  the  product  of  the 
heat  weight  imparted  in  a  certain  time  and  the  temperature  fall. 
Since  in  every  comj)lete  cycle  process  the  heat  weight  ab- 
stracted is  equal  to  that  imparted,  we  have  also 

L  =  -^j,iT~T,)    ....     (CXY.) 

Both  formulae  give  for  a  certain  Q  or  ^1  the  absolute  maxi- 
mum delivery  of  a  machine.  For  the  same  heat  Q  or  ^1  the 
delivery  is  greater  the  greater  the  temperature  fall.  In  hot-air 
engines  we  could  not,  on  account  of  practical  reasons,  have  T 
over  573°,  and  Ti  cannot  be  much  below  273°.  If  in  the  steam 
engine  T  were  573°,  or  t  =  300°,  we  should  have  an  enormous 
steam  pressure,  since  for  230°  the  pressure  is  about  28  atmos- 
pheres. At  present  we  seldom  exceed  10  atmospheres,*which 
corresponds  to  t  =  180.3°,  or  T  =  453.3°.  Whether  it  is  prac- 
ticable to  employ  higher  pressures  can  only  be  determined  by 
practice.  Further,  we  cannot  well  go  below  fi  =  46.2°,  or 
T,  =  319.2°,  as  this  temperature  corresponds  to  yV^h  of  an 
atmosphere.  For  a  lower  temperature  the  amount  of  condensa- 
tion water  is  too  great.  For  engines  without  condensation, 
^1  =  100°,  and  T^  =  373°.     If  then  we  regard  t  =  180.3°  as  the 


654  THERMODYNAMICS. 

maximum  temperature,  we  liave  for  the  maximum  delivery  of  a 
perfect  steam  engine,  for  condensation, 

=  ^||(180.3 -46.2)  =  125.38  g.     .     (CXVI.) 

and  non-condensing, 

L  =  ~||  (180.3  -  100)  =  75.08g  .     .     (CXVII.) 


EXAMPLE. 

What  is  the  delivery  of  a  perfect  steam  engine,  which  uses  per  hour  100  kilo- 
grams of  steam  of  10  atmospheres  ? 

If  we  assume  the  steam  to  be  dry,  Q  =  Gr  where  Q  and  G  are  quantities  per 
hour.  Now  r  =  p+  Apu  is  for  10  atmospheres  478.8.  Hence  Q  —  478.8  x  100 
=  47880  heat  units  per  hour,  or  13.3  per  second. 

For  a  condensing  engine,  then, 

L  =  125.38  X  13.3  =  1667.55  meter- kilograms, 


1667  55 
iV  =  — ^ —  =  22.23  horse  power. 

For  non-condensing, 

L  =  75.08  X  13.3  =  998.56  meter-kilograms, 

or, 

^^      998.56       ,oo«i 

JSr  =  — ^ —  =  13.32  horse  power. 

Since  now,  from  formulse  CXIY.  and  CXV.,  tlie  delivery  of 
an  engine  depends  only  upon  tlie  lieat  Q  and  temperature  fall, 
it  is  evident  tliat  the  kind  of  liquid  used  makes  no  difference, 
whether  water,  alcohol,  ether,  or  air. 

Let  us  now  consider  the  cycle  process  of  our  actual  steam 
engines  and  find  their  delivery.  We  shall  then  see  why  their 
cycle  process  is  not  perfect.  If  we  then  compare  the  delivery 
with  that  of  a  perfect  engine,  Ave  shall  have  the  loss  of  effect 
due  to  the  imperfection  of  the  process,  to  which  we  have 
already  referred. 


STEAM  ENGINE— CYCLE  PBOCESS. 


555 


C. — Cycle  Peocess  op  the  Actual  Steam  Engine  and  Deter- 
mination OF  the  Loss  op  Effect  due  to  the  Impeefection 
OP  the  Peocess. 


In  actual  engines  we  have  to  do  with  a  complete  but  not  a  per- 
fect cycle  process. 

Let  A  be  the  steam  cylinder  and  K  the  boiler.  The  steam 
has  the  press- 
ure p  and  tem- 
perature  t. 
Erom  the  boil- 
er it  passes 
through  the 
steam  pipe  to 
the  yalve  box 
on  the  right  of 
the  cylinder  A. 
Let  the  piston 
have  its  highest 
position,  a  n  d 
hence  the  up- 
per part  be  a 
little  open. 
Provided  that 
there  is  no  fric- 
tion in  steam 
pipe,  steam   of 

the  boiler  pressure  enters  above  the  piston  and  forces  it  down. 
Let  the  line  Oo  represent  the  pressure  p.  "When  the  piston 
has  passed  through  the  distance  0  V,  and  when,  therefore, 
there  have  entered  V  cubic  units  of  steam  from  the  boiler,  let 
expansion  commence,  and  the  steam  expand  according  to  the 
adiabatic  line  xx^.  At  the  end  of  expansion  let  the  steam  have 
the  pressure  pi  and  the  temperature  t^  of  the  condenser  C. 
Above  the  piston  we  have  then  steam  of  the  pressure  pi  and 
volume  Fi.  By  tliis  time  the  valve  has  opened  the  lower  port, 
and  steam  of  the  boiler  pressure  j:>  is  below  the  piston,  which 
now  rises.  While  rising,  it  forces  the  steam  volume  Fl  under 
the  pressure  jh  gradually  into  the  condenser.     Thus  the  back 


556  TEEBMODYNAMIGS. 

pressure  is  p^.  The  line  x/i  represents  this  pressure,  while  x-}> 
is  the  stroke. 

If  the  engine  has  a  surface  condenser,  the  volume  Fl  is  con- 
densed, and  is  then  forced  by  the  feed  pump  D  into  K.  Here 
it  is  again  heated  from  t^  to  t,  then  converted  into  steam,  and 
then  admitted  to  the  cylinder.  If  we  have  a  jet  condenser,  the 
pump  D  must  remove  from  the  condenser  not  only  the  con- 
densed steam,  but  also  the  injection  water.  But  it  only  has  to 
force  into  the  boiler  as  much  water  as  before.  If  there  is  no 
condenser,  the  back  pressure  px  is  that  of  the  atmosphere,  and 
it  is  just  the  same  as  if  we  had  a  condensing  engine  in  which 
the  condensed  steam  has  a  temjDerature  of  100''.  Although  in 
this  case  the  same  weight  of  water  must  be  forced  into  the 
boiler  as  before,  this  water,  if  there  is  no  feed  heater,  has  a 
lower  temperature,  and  more  heat  is  required  to  heat  it  than 
in  the  condensing  engine. 

Since  we  thus  know  the  character  of  the  cycle  process,  we 
can  calculate  the  delivery.  First,  it  is  evident  that  the  area 
hoxxi,  gives  the  delivery  per  stroke,  the  work  of  overcoming  the 
back  pressure  being  deducted  from  the  total.  If,  then,  we  de- 
duct the  work  required  for  forcing  the  feed  into  the  boiler,  we 
have  the  work  corresponding  to  the  cycle  process  of  the  actual 
steam  engine. 

We  assume  again  G  as  the  weight  of  steam  and  water  per 
stroke,  of  which  xG  kilograms  are  steam  and  (}.  —  x)  G  water. 
The  steam  volume  used  per  stroke  is  V,  or 

V  =  {xu  +  g)  G  cubic  meters. 

The  work  during  full  pressure  is 

Li  =  2^V = 2^ i^^^  +  ^)  G-  meter-kilograms. 

If  the  specific  steam  weight  at  the  end  of  expansion  is  Xi,  we 
have  for  the  volume  V^ 

Vx  =  {x,ut  +  (t)  G,. 

Since  the  steam  expands  adiabatically,  we  have  for  the  work 
during  expansion  (Equation  XXVII.) 

C 

L^  =  {q-qi  +  xp-  x,p,)  -j . 


STEAM  ENGINE— CYCLE  PROCESS.  557 

Hence  tlie  cTeliyeiy  per  stroke  is 

Li  +  7.2  =  [p  [o::u  +  ff)  +  -^  (q  —  qi  +  xp  —  x^p-i}]  G. 

From  this  we  must  subtract  the  work  in  overcoming  the  back 
pressure  px.     This  is 

Xg  =  Px   {X{Ul    +    (j)     G. 

Therefore  the  delivery,  neglecting  the  work  required  for  the 
feed,  is 

A  {Li  +  X2  —  X3)  =  \_Ap  {xu  +  ff)  +  {q  —  qi  +  xp—  x^p-^  —  Aih 
{xiUx  +  c)]  G  heat  units. 

G 
Li  +  L2  —  L3  =  L  —  [q  —  qx  +  xr  —  XxVx  +  (j  {p  —p^  ^]  "t  '  °^ 

AL  =\ci  —qx  +  xr—  XxVx  +  A(f  Q:*  —  pi)]6^  heat  units. 

(CXYIIL) 

If  the  last  member  in  this  equation  is  neglected  on  account 
of  its  smallness,  we  shall  have  Equation  LY.,  j)age  474,  which 
we  have  already  found  for  the  efflux  of  steam.     Hence 

A~G  =  AL, 

2cj 


2g 

That  is,  the  total  delivery  of  the  actual  steam  engine  per 
stroke  is  equal  to  the  living  force  of  the  steam  G  used  per 
stroke  when  it  flows  with  the  velocity  tv  from  the  boiler.  This 
might  at  first  sight  seem  to  make  it  advantageous  to  allow  the 
steam  to  act  by  impulse  or  reaction.  "When  we  consider,  how- 
ever, that  a  reaction  wheel  only  gives  its  maximum  delivery 
when  it  revolves  with  the  same  velocity  as  the  liquid  departs, 
such  a  wheel  would  have  to  have  an  enormous  velocity,  as  the 
velocity  of  steam  is  very  great  when  issuing  even  under  low 


558  THERMODYNAMICS. 

pressures.  Such  a  Telocity  would  consume  much  of  the  effect, 
even  if  the  construction  had  any  practical  value.  (Zeuner, 
Warmetheorie,  page  477.) 

If  we  determine  x{t\  from  the  known  relation 

xr  X]i\ 

and  insert  it  in  the  preceding  equation,  we  have,  after  reduc- 
tion, 

AL^\^^{T-T,)^-q-q,-{r-r,)T,  +  A6{p-p,)]^G.  (CXIX.) 


This  expression  is  not  yet  the  outer  work  of  the  steam  en- 
gine. It  is  rather  the  entire  work  obtained  up  to  the  point 
where  the  G  kilograms  of  steam  at  the  temperature  t  have  be- 
come water  at  the  temperature  t^.  To  complete  the  cycle  pro- 
cess we  have  still  to  force  this  water  into  the  boiler. 

Here  we  have  to  distinguish  between  engines  condensing  and 
non-condensing.  In  the  first  case,  we  may  have  either  a  sur- 
face or  a  Jet  condenser.  In  the  other,  the  steam  escapes  into 
the  air. 

(«.)  Condensing  Engine — Surface  Condenser. — In  this  case 
the  pump  D  has  to  raise  per  stroke  the  G  kilograms  and  force 
them  into  the  boiler.  If  the  pressure  of  the  air  is  p^,,  we  have 
for  the  work  of  removing  from  the  condenser 

and  for  forcing  into  the  boiler 

Gg{p-Pq). 
When  we  add  both  works,  we  have  for  the  work  of  the  pump 
Li^=  Oa  {p  —  pi)  meter-kilograms,  or  in  heat  units 
AL^  =  Ag  {p  —2h)  G- 


STEAM  ENGINE— SURFACE  CONBENSEB.  559 

Subtracting  tliis  from  CXIX.,  we  liaye  for  tlie  work  obtained 
by  the  cycle  process  of  a  steam  engine  with  surface  condenser, 

AL=  r^\r-rO  +  ^-gi-(r-ri)^i1  6^  heat  units.      (CXX.) 

(5.)  Condensing  Engine — Jet  Condenser. — "We  suppose  that  the 
air  pump  not  only  removes  the  water  from  the  condenser,  but 
also  forces  it  into  the  boiler.  We  also  neglect  the  fact  that  it 
removes  air  also.  If  (zq  is  the  injection  water  per  stroke,  the 
pump  has  to  remove  per  stroke  G  +  Gq,  but  only  has  to  force 
G  into  the  boiler.     The  work  of  removing  is 

{Go  +  G)  (po  -pi)(r=  Go(T{po  ~pi)  +  Ga  {po  -p,). 

The  water  (?„  then  runs  off,  and  G  is  forced  into  the  boiler. 
The  work  required  is 

Gff{p-2:)o). 
The  total  work  of  the  pump  is  then 

A  =  Goff  {p,  -  p,)  +  Ga{p-  p,). 

We  have  then  for  the  work  of  the  cycle  process  of  a  condens- 
ing engine  with  jet  condenser 

AL^  [|;  {T-  T,)+q-q,-{r-r,)  T^l  G  -  G,a  {p,  -  p,)  A. 

(CXXI.) 
The  last  member  is  so  small  that  it  may  be  neglected. 

(c)  Non-condensing  Engine,  zvitJi  Ordinary  Force  P^cmp. — Let 
the  height  to  which  the  water  is  sucked  be  Ji.  The  work  is 
Gh.     The  work  of  forcing  into  the  boiler  is 

G(r{p-po). 

But  in  the  present  case  po  =  p^,  hence 

Li  =  Gh  +  Gff  {p  —pi)^ 


560  THERMODYNAMICS. 

Subtracting  tliis  work  from  CXIX.,  we  have 

AJ=  [I'  {T-T,)  +  q-q,-  (r-rO  T,-  ^a].     (CXXIL) 

Here  Ah  may  be  neglected,  and  we  have  again  Equation  CXX. 

(d)  The  Boiler  is  Fed  by  a  Giffard  Injector. — ^In  this  case 
there  is  no  outer  work  required  for  the  injector.  We  can 
therefore  use  Equation  CXIX.  directly,  of  which  we  may  ne- 
glect the  last  equation.  Hence  Equation  CXX.  gives,  in  all 
cases,  the  total  delivery  measured  in  heat  units  of  the  cycle 
process  of  the  steam  engine. 

The  question  now  arises,  what  is  the  amount  of  heat  ex- 
pended? If  we  know  this,  and  insert  it  in  Equation  CXIY. 
instead  of  Q,  we  shall  have  the  delivery  of  an  engine  with  per- 
fect cycle  process.  The  comparison  of  this  with  Equation 
CXX.  will  give  the  loss  of  effect  by  reason  of  the  imperfection 
of  the  process.     This  heat  can  be  easily  determined. 

We  assume  first  that  the  engine  is  condensing.  The  G  kilo- 
grams of  steam  used  per  stroke  are  removed  in  liquid  state 
from  the  condenser  and  forced  into  the  boiler.  The  tempera- 
ture of  this  water  we  have  indicated  by  t,  and  the  heat  of  the 
liquid  is  qi.  Hence  the  G  kilograms  of  water  contain  Gqi  heat 
units.  In  the  boiler  the  temperature  is  raised  to  f,  and  the 
heat  of  the  liquid  is  q.  The  heat  imparted  is  then  G{q  —  q-i). 
Of  this  water  xG  kilograms  are  now  vaporized,  which  requires 
the  heat  Grx. 

The  total  heat  then  is 

G  (rx  +  q  —  q^)  =G[q  —  qi  +  {p  +  Apicjx]. 

Let  us  assume  again  that  the  engine  is  non-condensing,  and 
that  the  feed  is  furnished  by  the  injector.  If  the  feed  water 
has  a  temperature  of  4  the  heat  of  the  liquid  is  qc,  and  if  we 
assume  the  water  heated  to  t^°  by  the  steam  of  the  injector, 
this  steam  must  itself  lose  heat,  so  that  it  becomes  water  at  ^i°. 
When  now  this  condensed  steam  with  the  feed  water  enters  the 
boiler,  the  former  must  receive  as  much  heat  as  it  lost  in  con- 
densing: in  order  to  be  converted  into  steam  of  f.     We  neglect 


STEAM  ENGINE— CYCLE  PROCESS.  561 

thus,  indeed,  that  heat  which,  transformed  into  work,  is  neces- 
sary to  raise  the  feed  water  and  force  it  into  the  boiler.  On 
account  of  its  slight  comparative  amount,  this  is  allowable.  If 
therefore  the  quantity  of  water  raised  is  G,  and  if  it  is  heated 
to  ^1°,  the  heat  lost  by  the  steam  in  thus  heating  it  is 

^  (^1  -  <Zo), 

and  this,  as  remarked,  must  be  again  imparted  to  the  condensed 
steam  in  the  boiler  in  order  to  convert  it  into  steam  at  t°.  The 
feed  water  is  now  heated  in  the  boiler  from  ^i"  to  i,  and  this  re- 
quires the  heat  G  {q  —  q,).  The  total  heat  imparted  to  the  feed 
water  in  order  to  bring  it  up  to  the  temperature  of  the  boiler 
water  is  then 

Gq,  -  Gqo  +  Gq-Gq,  =  G{q-  q,). 

Finally  this  water  is  to  be  converted  into  steam  at  f.  For 
this  we  require,  assuming  that  of  the  total  weight  of  feed  water, 
Gx  are  steam,  Grx  heat  units.  Hence  the  total  heat  imparted 
to  the  feed  water  is 

G  [rx  +  q  —  qo)  =  G[q  —  q^  +  (p  +  Apu)  a*]  heat  units. 

If,  finally,  the  boiler  is  fed  by  an  ordinary  force  pump,  and 
the  temperature  of  the  feed  water  is  ^o?  we  have  again 

G  {rx  +  q  —  qi))  =  G  [q  —  qo  i-  {P  +  Apu)  x]  heat  units, 

in  order  to  form  steam  of  f°.  We  see,  then,  that  with  the  in- 
jector we  have  to  impart  as  much  heat  to  the  feed  water,  in 
order  to  generate  the  steam  required,  as  when  the  ordinary 
force  pump  is  used.  In  this  respect  also,  then,  the  injector 
possesses  no  advantage.  The  single  advantage  of  the  injector 
is  that  the  frictional  resistances  are  less. 

From  the  above  it  follows  that  the  heat  required  for  the  de- 
livery of  the  cycle  process  of  the  steam  engine  is  given  by 

Qi  =  G  {rx  +  q  -  q^)  =  G[q  -  qo  +  {p  +  Apu)  x], 

in  which  for  condensing  engines  we  put  q^  for  go- 
36 


562  THERMODYNAMICS. 

Tlie  heat,  then,  is  less  the  greater  g,-,  that  is,  the  hotter  the 
feed  water.  We  see,  then,  the  desirableness  of  a  feed  water 
heater. 

If  now  this  heat  is  used  in  a  perfect  engine,  we  have  for  the 
delivery 

This  delivery  is  therefore  that  which,  from  the  standpoint 
of  the  mechanical  heat  theory,  is  disposable  when  the  heat 
Gr  {q  —  go  +  fx)  is  used.  It  is  more  convenient  to  ]3ut  the  for- 
mula in  the  following  form, 

z„=.g['|(r-r.)  +  ^^-^"^f-^'^].   .   (cxxiii.) 

Now  in  our  actual  steam  engines  we  have  the  work  from 
Equation  CXX., 


J.       G  frx 
^  =  A 


\^-^{T-T,)+q-q,-{r-r,)T,'J. 


If  we  subtract  this  from  the  preceding,  we  have  for  the  loss 
of  work  by  reason  of  the  imperfection  of  the  cycle  process, 

A  -  ^  [(</!  -qo)T-(q-  go)  T,  +  {r-  r,)  TT,].  (CXXIY.) 

For  condensing  engines  we  put  qi  in  place  of  q^.  If  we  divide 
this  by  the  disposable  work,  we  have  the  ratio  of  the  loss  of 
effect  to  the  work  which  is  at  our  disposition  in  the  heat  used. 
This  ratio  is 

,,  _  (gi  -</o)  T-{q-  go)  T,  +  {r-  r,)  TT, 


A  non-condensing  steam  engine  works  with  dry  steam  of  5  atmospheres.  "What 
is  w  when  the  feed  water  has  a  temperature  t„  =  15°  ? 


5 

atmospheres 

t  =  153.2 

q  =  153.7 

r  =  0.45 

r  =  499.3 

STEAM  ENOINE—LOSS  IN  CYCLE  PROCESS.  563 

From  our  Tables  we  ha^-e  for 

and  for        1  atmosphere 

t,   =100 
q^  =  100.5 
r^  =r  0.31 
r  =  499.3 
hence 

_  (100.5  -  15)  435.3  -  (153.7  -  15)  373  +  (0.45  -  0.31)  425.3  x  373 
^*'  ~  (153.7  -  15  +  499.2)  52.3  ~ 

6005.33  _ 
=  33T9"8:4-^-^^- 

Therefore  18  per  cent,  of  the  work  at  disposal  is  lost  by  reason  of  the  imper- 
fection of  the  cycle  process. 

Zeuner,  to  wliom  tliis  elegant  and  interesting  discussion  is 
due,  has  investigated  by  various  examples  the  influences  of 
heating  the  feed  water,  and  of  water  contained  in  the  steam, 
upon  the  loss  of  effect.  The  following  tabulation  in  which  the 
steam  is  assumed  to  be  dry,  shows  the  influence  of  heating  the 
feed  water.     The  engines  are,  of  course,  non-condensing. 

Non-Condensing  Engines. 
{Back  Pressure,  1  Atmosphere.) 


Boiler  Presi^nre 

in 

Atmospheres 

Loss  of  Effect 
for 

?,  =  15»           and 

IV 

H 

0.15 

0.01 

3 

0.16 

0.03 

4 

0.17 

0.04 

5 

0.17 

0.05 

6 

0.18 

0.05 

8 

0.19 

0.06 

10 

0.19 

0.07 

If,  then,  the  feed  water  is  heated  up  to  100",  the  loss  of 
effect  by  reason  of  the  imperfection  of  the  cycle  process  is 
small,  especially  for  low  steam  pressures.  For  condensing  en- 
gines, in  which  the  back  pressure  is  about  yV  of  an  atmosphere, 
in  which,  therefore,  the  feed-water  temperature  is  t^  =  46.2°, 


564  THERMODYNAMICS. 

the  loss  for  li  atmospheres  is  0.05,  and  for  10  atmospheres 
0.10.  Here,  then,  the  loss  of  effect  increases  with  the  pressure. 
In  the  following  tabulation  we  see  that  the  loss  of  effect  in- 
creases with  the  quantity  of  u'ater  in  the  steam.  The  engine  is 
assumed  non-condensing  and  working  with  steam  of  5  atmos- 
pheres. 

Specific  steam  Loss  of  ellect  for  feed  water  temperatures, 

quantity.  f^  ^  I50  gQO  -j^qqo 

x  =  l  IV  =  0.17  0.08  0.05 

x  =  0.90  0.19  0.09  0.05 

a^  =  0.80  0.21  0.10  0.06 

We  see  from  both  tabulations  how  advantageous  it  is  to  use 
hot  feed  water.  This  is  confirmed  by  experiment.  The  waste 
gases  in  the  chimney  may  be  used  for  heating  the  feed.  But 
even  then,  the  loss  of  effect  for  high  pressures,  especially  when 
the  steam  contains  10  or  20  per  cent,  of  water,  is  considerable, 
so  that  it  becomes  a  question  whether  the  cycle  process  of  our 
present  steam  engines  can  be  so  altered  as  to  correspond  to 
that  of  a  perfect  steam  engine.  For  this  purpose,  we  should 
evidently  not  condense  all  the  steam  in  the  condenser,  but 
rather  so  much  should  remain,  that  by  adiabatic  compression 
this  remaining  steam  may  be  converted  into  water,  with  the 
already  condensed  steam,  at  the  boiler  temperature  {t).  Never- 
theless, the  preceding  discussion  shows  that  on  the  whole  the 
imperfection  of  the  process  is  small.  The  other  losses,  as  that 
due  to  imperfect  expansion,  prejudicial  space,  etc.,  are  in  part 
greater,  at  least  the  loss  due  to  imperfect  cycle  is  but  a  small 
part  of  the  total  losses.  Accordingly,  it  is  by  no  means  correct 
as  Redtenbacher  asserts,  that  the  cycle  process  of  our  steam 
engines  is  exceedingly  imperfect,  and  that  therefore  some  other 
method  of  utilizing  the  steam  should  be  invented.  "  So  long 
as  the  fundamental  principles  of  the  mechanical  theory  of  heat," 
says  Zeuner,  "  are  regarded  as  correct,  so  long  we  can  regard 
the  cycle  process  of  our  steam  engines  as  quite  perfect,  and  if 
there  are  no  losses  of  work  greater  than  that  due  to  the  essen- 
tial imperfection  of  the  process,  we  need  not  search  for  im- 
provement in  the  steam  engine  ;  at  any  rate,  in  those  engines 
which  use  saturated  steam." 


CHAPTER  XXV. 

COMPLETE   CALCULATION   OF   THE   STEAM  ENGINE. 

Indicated  Delivery. — We  have  in  tlie  preceding  calculated  the 
total  delivery,  or  that  obtained  by  the  cycle  process  of  the  or- 
dinary steam  engine,  when  we  disregard  the  work  required  for 
the  feed.  TVe  have  now  to  determine  more  exactly  the  work  of 
the  steam  in  the  cylinder,  not  only  with  reference  to  the  back 
pressure,  but  also  to  other  prejudicial  actions.  We  have  first 
to  calculate  the  work  which  the  steam  actually  performs  on  the 
piston.  Since  this  work  is  accurately  given  by  the  indicator 
diagram,  we  call  it  the  indicated  delivery  or  horse  power  of  the 
steam  or  engine.  In  these  calculations  Ave  shall  proceed,  of 
course,  from  the  principles  of  the  mechanical  heat  theory. 
Then  we  shall  show  how  the  prejudicial  resistances,  such  as 
piston  and  valve  friction,  that  of  fly-wheel  and  pumps,  etc.,  are 
to  be  determined.  Finally,  we  shall  show  how  to  determine 
the  dimensions,  the  consumption  of  fuel,  etc.,  for  an  engine  of 
given  horse  power. 

Let  us  first  examine  more  closely  the  action  of  the  steam  in 
the  cylinder.  We  assume  an  engine  with  ordinary  slide  valve, 
moved  by  an  eccentric. 

Action  of  the  Steam  in  the  Cylinder. — In  Figs.  94,  95,  96,  97,  and 
98,  AB  is  the  cylinder,  CB  the  piston,  EF  the  slide  valve  box, 
/  a  portion  of  the  steam  pipe,  GH  the  slide  valve,  ah  and  cd 
the  steam  passages,  a  and  c  the  entrance  ports,  and  e  the  ex- 
haust port,  through  which  the  steam  passes  either  into  the  air 
or  into  the  condenser.  In  Fig.  94,  the  piston  is  at  the  left  fend 
of  its  stroke ;  the  port  a  is  already  a  little  open,  and  steam  enters 
from  the  boiler  and  presses  upon  the  left  side  of  the  piston. 
This  steam  we  shall  call  the  ''■driving  steam."     This  opening  of 

5G5 


566 


THEBMOD  TNAMIC8. 


the  entrance  port  for  the  admission  of  steam,  "before  the  piston 
gets  to  the  end  of  its  stroke,  is  necessary  for  smooth  mo- 
tion of  the  engine. 
While  now  the  driving 
steam  forces  the  pis- 
ton to  the  right,  the 
yalve  moves  also  in 
the  same  direction, 
and  the  port  a  is 
opened  more  and 
more,  and  fresh  steam 
a  continually  enters. 
Finally  the  port  a  is 
fully  opened,  and  the 
valve  has  then  reached 
its  extreme  position 
towards  the  right.     It 

Fig.  94.  .,  i         •  j. 

then  begins  to  move 
towards  the  left,  and  thus  closes  a  more  and  more,  so  that  the 
steam  enters  with  increasing  resistance.  In  Fig.  95,  the  port  a 
is  completely  closed,  and  hence  no  more  steam  can  enter  behind 
the  piston.     Since' the  piston  has  not  yet  arrived  at  the  end  of  its 


y//77777///y/////////777//////////y777Amm. 
Fig.  95. 


stroke,  however,  the  driving  steam  must  now  act  expansively. 
Meanwhile  the  valve  still  goes  towards  the  left,  and  in  Fig. 


8TEA3I  ENGINE— ACTION  OF  STEAM. 


567 


96,  we  haye  tlie  position  of  valve  and  piston  when  tlie  first  is 
about  to  open  tlie  port  a  for  tlie  discliarge  of  the  driving  steam, 
while  the 
second  is  not 
yet  at  the  end 
of  its  stroke 
towards  the 
right.  Up  to 
this  moment 
we  have  ex- 
pansion o  f 
the  driving 
steam,  but  of  | 
course  not 
after.  This  is 
then      the 

.     ,         „  Fig.  96. 

point  oi  re- 
lease, while  Fig.  95  is  the  position  where  expansion  begins. 
From  this  point  on,  the  valve  opens  the  port  a  for  the  release 
of  the  steam.  The  driving  steam  flows  through  a,  h,  and  e  to 
the  condenser,  or  out  into  the  air.  Fig.  97  shows  the  position 
of  piston  and  valve  when  the  port  a  is  tolerably  open  for  dis- 
charge, while  the  piston  has  not  yet  arrived  at  the  right  end  of 
its  stroke. 

In  this  position  the  port  c  is  closed.     While  now  the  piston 

still  moves  to- 
E        f^^^a  I         F  ward  the  right 

H  and    the   valve 

A  ^^fe^^  ^^^Sj^J         Hvi  toward  the  left, 

^^"  r///.)  -  m^.^m:^22zM  tl,e    port    c  is 

opened  to  ad- 
mit steam,  and 
the  steam  en- 
ters as  before 
it  did  iu  Fig. 
94,  only  on  the 
right  side  of  the 
piston  instead 
of  the  left.  The 
driving  steam  is  now  on  the  right  of  the  piston.     Thus  far  we 


568 


THEBMOD  TNAMIC8. 


have  confined  our  attention  to  tlie  left  side  of  the  piston  and 
have  considered  the  action  of  the  driving  steam.  Let  ns  now 
see  how  the  back  pressure  steam  acts.  This  steam  we  have  in- 
dicated by  points  in  our  Figures,  while  the  driving  steam  is 
indicated  by  horizontal  lines.  Let  us  refer  again  to  Fig.  94 
We  see  here  that  the  port  c  is  tolerably  wide  open,  wider  than 
a,  and  hence  that  the  release  of  the  back  pressure  steam  to  the 
condenser  takes  place  before  the  admission  of  the  driving  steam. 
From  this  point,  as  the  valve  moves  to  the  right,  c  is  opened  more. 
It  is  fully  opened  when  the  slide  is  at  its  extreme  right  posi- 
tion. Fig.  96  shows  the  valve  returned  a  good  ways  toward  the 
left,  but  still  in  communication  with  the  condenser.  The  steam 
in  the  cylinder,  on  right  of  piston,  has  then  the  pressure  of  the 
atmosphere,  or  of  the  condenser.  In  Fig.  97  the  port  c  is  closed, 
and  as  the  piston  still  goes  toward  the  right,  the  steam  is  com- 
pressed, becomes  hotter,  denser,  and  has  a  higher  pressure. 
For  this  compression  a  certain  work  is  necessary,  which  must 
be  deducted  from  the  total  delivery.  But  this  work  is  not  lost, 
since  now  less  fresh  steam  is  required  for  filling  the  space  back 
of  the  piston.  Experience  shows  that  this  compression,  or 
"  cushioning,"  is  also  necessary  for  quiet  and  smooth  working. 
Fig.  98  shows  the  end  of  compression,  or  the  position  of  the 
valve  when  c  just  begins  to  open  for  the  admission  of  fresh 

steam  on  the  right. 
The  action  of  the 
steam  just  described 
is  caused  by  two 
things,  the  ^^  angle  of 
advance"  and  the 
'Hap"  of  the  slide 
valve. 

At  first,  things  were 
so  arranged  that  both 
ports  were  closed 
when  the  engine  was 
on  its  dead  points  ;  in 
other  words,  when  the 
■^"^■^^"  piston  was  at   either 

end  of  its  stroke. 

The  slide  was  then  in  its  central  position.     Such  a  relation 


STEAM  ENGINE-ACTION  OF  STEAM. 


669 


between  tlie  motion  of  tlie  slide  and  piston  can  be  easily 
attained  by  so  placing  the  eccentric  disc  that  the  line  joining 
its  center  with  the  center  of  the  shaft  makes  an  angle  of  90° 
with  the  line  joining  the  dead  points.  If  thus  B  and  C, 
Fig.  99,  are  the 
dead  points  for 
a  horizontal  cy- 
linder, the  ec- 
centric   disc   8  _    ^ 

must     be    so  ^ 

placed    on    the  ^^^  ^g 

shaft    D,     that 

Z^-f/ makes  an  angle  of  90°  witlii>(7.  In  this  case  both  ports 
would  be  fully  opened  when  the  piston  is  in  the  middle  of  its 
stroke,  if  the  connecting  rod  were  infinitely  long.  Even  for 
moderate  length  of  this  rod,  the  same  is  nearly  true.  Since 
now,  in  our  present  arrangement,  we  wish  both  ports  to  be 
open  when  the  piston  is  at  either  end  of  its  stroke,  the  valve 
must  be  beyond  its  central  position.  This  is  attained  by  fixing 
the  disc  on  the  shaft  so  that  DE  makes  more  than  90°  with  BC. 
This  increase  of  the  angle  of  90°  is  the  "  angle  of  advance." 
The  angle  of  advance,  then,  is  the  angle  made  by  the  eccen- 
tricity with  the  perpendicular  to  the  valve  face  when  the  pis- 
ton is  at  a  dead  point. 

If  now  the  steam  is  required  to  act  in  the  cylinder  with  a 
certain  expansion  and  compression,  we  must  have  the  following 


arrangement.  We  make  the  slide  so  long  that  in  its  central 
position  it  laps  over  the  ports  by  a  certain  amotint  on  each 
side.  The  amount  which  it  exceeds  the  port  on  the  outside  is 
called  the  "outside  lap,"  and  on  the  inside,  the  " inside  lap.'' 
As  it  is  more  advantageous,  as  has  been  pointed  out,  for  the  re- 


570 


THERMOD  YNAMICS. 


lease  of  the  steam  to  take  place  somewhat  earlier  than  the 
admission,  the  outer  lap  is  always  greater  than  the  inner. 
Thus  ah  and  cd,  Fig.  100,  are  the  outer  laps,  and  ef  and  gh  the 
inner,  or  they  are  the  distances  by  which  the  valve,  when  in 
its  central  position,  extends  beyond  the  ports. 

For  this  central  position  of  the  valve  the  center  d  of  the  ec- 
centric disc  must  be  in  the  perpendicular  erf  to  he.  If,  now, 
when  the  piston  is  at  the  left  end  of  its  stroke,  the  port  for  the 


-i)-ii- 


admission  of  steam  on  the  left  is  to  be  opened  a  little,  the  point 
d  must  be  somewhere  to  the  right,  say  at/.  If  the  conecting 
rod  and  eccentric  rods  are  very  long,  the  travel  of  the  valve 
toward  the  right  is  approximately//. 

The  angle  clef  is  the  angle  of  advance.  If  we  wish  to  know 
how  far  the  piston  is  from  the  left  end  of  its  stroke,  when  the 
valve  has  its  central  position,  we  have  only  to  lay  off  the  angle 
hell  =  dcf,  and  let  fall  Id.  Then  hi  is  the  distance,  if  he  =  ce  is 
the  angle  of  the  crank,  or  the  half  stroke  of  the  piston. 

Let  the  .angle  of  advance  def  =  a,  and  the  eccentricity  ed, 
that  is  the  distance  of  the  eccentric  disc  from  the  center  of  the 
shaft  be  p,  then 

of  —  p  sin  c\ 


STEAM  ENGINE— ACTION  OF  STEAM.  57I 

If,  now,  the  outer  lap  ab  —  cd  —  «i,  and  the  "  lead,"''  or  the 
opening  of  the  port,  when  the  piston  is  at  the  end  of  its  stroke, 
is  hi,  we  have 

gf  =  CI1  +  hi  =  p  sin  a'. 

In  general,  for  horizontal  engines,  cii  =  0.25p,  or  ^  of  the  ec- 
centricity, and  bi  =  -^^p,  hence 

(0.25  +  0.1)  p  =  p  sin  a, 
or 

sin  a  =  0.35,     or     a  =  20°  30'. 

If  further,  the  inner  lap  ef  =  gh  is  a^,  the  opening  of  the  port, 
when  the  piston  is  at  the  end  of  its  stroke,  for  the  release  of 
the  steam,  or  the  inside  lead,  is  h^,  we  have 

a^  +  1)2  —  p  sin  a  —  0.35p. 

If  we  make  the  inside  lap  a^  =  0.05p,  that  is  -^th  of  the  out- 
side lap,  we  have 

O.OSp  +  &o  ^  0.35p,     or     b^  =  O.SOp. 

Hence  the  port  is  opened  for  discharge  three  times  as  much 
as  for  entrance,  when  the  piston  is  on  dead  point. 

We  can  now  easily  find  the  angle  a\  or  cu  through  which  the 
center  of  the  eccentric  disc  must  turn,  in  order  to  open  the 
port  for  entrance  or  discharge.     For  the  first, 

p  sm  n\  =  a-i,     or     sm  a\=  — , 

or  inserting  value  of  «i,  viz.,  0.25p, 

sin  a;  =  0.25,     or     n',  =  14°  29'. 

.                               .eta      0.05/3 
Jjurther,  p  sm  oo  =  «25     or     sm  «o  =  —  —  ■ =  0.5,   or 

p  p 

a^  =  2°  52'. 

The  eccentric  has  to  turn  but  a  little,  therefore,  from  its  cen- 
tral position,  in  order  to  open  the  port  for  discharge. 


572  THERMODYNAMICS. 

If  the  piston  is  at  the  left  end  of  its  stroke,  the  end  of  the 
crank  is  at  &,  or  at  one  dead  point.  If  now  h  passes  through 
the  arc  hlz,  we  can  easily  find  the  travel  of  the  piston.  Since 
the  connecting  rod  is  very  long  compared  to  the  eccentricity, 
this  travel  is  hi,  and  since  cb  is  half  the  stroke,  or  \  s, 

hl  =  hc  —  cl  =  -^  —'^  cos  cp  =  '--  (i  —  cos  (p). 


For  9?  =  90^,  cos  cp  =  0  and  the  travel  is  -^.     For  9  =  180°, 

A 

cos  q)  —  ~1,  and  ^  (1  —  (  —  1))  =  s.     Generally  the  travel  of 
A 

the  piston  x  for  any  angle  of  the  crank  is 

X  =  -^  {1  —  cos  qj). 
A 

When  the  crank  is  turned  through  the  arc  cp  from  its  dead 
point,  the  eccentricity  makes  the  angle  a  +  qj  with  dc,  and  the 
travel  10  of  the  valve  is 

10  —  p  sin  [a  +  q)). 

This  travel  is  positive  when  the  motion  of  the  slide  is  in  the 
same  direction  as  that  of  the  piston,  otherwise  it  is  negative. 

1.  Travel  of  the  Piston  tip  to  the  End  of  Admission  or  to  the 
Beginning  of  Expansion. — Let  this  travel  s^  be  a  portion  ei  of  s, 
so  that 

5i  =  ejs. 

We  wish  to  find  e^.  The  entrance  of  steam  ends  when  the  end 
a  of  the  valve  returns  to  b  again,  on  the  back  stroke.  Before 
the  port  begins  to  open,  the  valve  must  pass  through  the  dis- 
tance ah  =  aj,  or  the  eccentricity  through  the  angle  den  =  a\. 
If  we  let  fall  from  u  a  perpendicular  to  he  and  prolong  it  to  0, 
00  is  the  position  of  the  eccentricity  when  the  valve  again  closes 
the  port.  The  eccentric  revolves  then,  from  beginning  of 
admission  to  cut-off,  through  npo  =  180  —  2fl'i.  The  crank,  of 
course,  goes  through  the  same  angle.  But  when  the  eccen- 
tricity is  at  n,  the  crank  makes  an  angle  with  he  of  a'  —  a\.     If 


STEAM  ENGINE— ACTION  OF  STEAM.  573 

•we  denote  the  rest  of  the  arc  tlirotigli  whicli  the  crank  must  go 
by  y,  we  haye 

a  —  a\  -\-  y  =  180  —  Srv^,     or 

2/  =  180  -  (rtr  +  n-i)- 
If  we  insert  this  in  the  equation  of  the  travel  of  the  piston, 


ei-s  =  —  (1  +  cos  {a  +  a^),  hence 
A 

_  1  +  cos  {a  +  n'l) 


Since  a  =  20°  30',  a^  =  14°  29',  we  have  e^  =  0.910.  The  ad- 
mission of  steam,  therefore,  ceases  when  the  piston  has  passed 
through  0.91  of  its  stroke.     Expansion  then  begins. 

2.  Travel  of  the  Piston  up  to  tlie  Beginning  of  Compression,  or 
the  End  of  Belease. — The  compression  begins  when,  on  the  re- 
turn of  the  valve,  the  corner  g  meets  h.  The  valve  is  then  dis- 
tant gh  from  its  centre  position,  and  the  eccentricity  makes  the 
angle  (^  with  cq.  From  d  then  it  makes  the  angle  180  —  o'^. 
The  crank  has  passed  through  the  same  angle  from  h  or  from 
h  through 

180  -a'^-a  =  180  -{a  +  a'^). 

But   ■  cos  [180  —  {a  +  a'2)]  =  —  cos  {a  +  o'g), 

and  denoting  the  travel  up  to  beginning  of  compression  by 

S3  —  e^s, 

S3  =  egS  =  a?  =  -  [1  +  cos  (a  +  a^)],  or 
A 


__  1  +  COS  (a  +  a^ 


2 


If  we  put  for  a  and  a^  the  numerical  values, 

63  =  0.959. 
This  position  of  the  piston  is  shown  in  Fig-  97. 


574  THEBM0BTNAMIG8. 

3.  Travel  of  the  Piston  up  to  the  End  of  Expansion  or  to  the 
Point  of  Release. — The  expansion  ends  wlien  the  valve  has 
moved  so  far  towards  the  left  that  /  coincides  with  e.  The 
valve  has  then  moved  towards  the  left  ef  from  its  central  posi- 
tion. Since  ef  corresponds  to  the  angle  a^,  the  eccentricity 
makes  this  angle  to  the  left  of  q.  Reckoned  from  d,  the  angle 
is  180  +  a'2.  The  crank  has  made  the  same  angle,  reckoned 
from  h,  or  from  h  it  makes 

180  -{-  a^-a  =  180  -{a-  a^). 
If  the  travel  in  the  present  case  Sg  =  e^s,  we  have 

Si  ^e^  =  ^\l  +  cos  {a  —  n-g)], 

and 

_  1  +  cos  {a  —  o'a) 
e,  -  2  • 

Inserting  the  numerical  values, 

e,  =  0.977. 

Since  expansion  begins  at  0.910,  the  duration  of  expansion  is 
0.977  -  0.910  ^  0.067  of  the  entire  stroke. 

The  position  of  valve  and  piston  in  this  case  is  shown  in 
Fig.  96. 

4.  Travel  of  the  Piston  up  to  tJie  End  of  Compression  of  the  Bach 
Pressure  Steam,  or  up  to  the  Admission  of  Driving  Steam. — The 
compression  ends  when  the  point  d  of  the  valve  coincides  with 
c.  The  valve  has  then  moved  cd  from  its  central  position  to- 
wards the  left.  The  eccentricity  makes  the  angle  a\  with  q,  or 
180  +  a\  with  d,  hence  the  crank  makes  from  h  the  angle 

180  +  a'l  -  or  =  180  -{a-  a^). 

The  travel  of  the  piston  up  to  the  end  of  compression  is  then 

Si  =  64.9  =  cc  =  ^  [1  +  cos  {a  —  a'l)],  or 

_  1  +  cos  {a  —  n'l) 
64-  ^  . 


STEAM  ENGINE— ACTION  OF  STEAM.  575 

Inserting  tlie  numerical  values,  64  =  0.997. 

The  piston  lias  then  only  a  very  small  distance  to  go  to  its 
dead  point.     This  position  is  shown  In  Fig.  98. 

Since  for  the  beginning  of  compression  e^  =  0.959,  and  for  the 
end  ^4  =  0.997,  the  entire  compression  is 

0.997  -  0.959  =  0.038  of  the  stroke,  or  about 

half  as  great  as  the  expansion.  We  see  also  that  the  compres- 
sion begins  later  and  ends  later  than  the  expansion.  It  is  in- 
dispensable for  smooth  motion  of  the  engine. 

The  fraction  e^  of  the  entire  stroke  s,  which  is  filled  with 
boiler  steam,  is  the  "  coefficient  of  fill."  For  the  usual  propor- 
tions as  given,  it  is  0.91.  This  value  may  be  regarded  as  a 
maximum.  If  the  engine  has  a  special  expansion  valve,  ei  is 
less,  but  the  other  quantities  e..,  e^,  and  e^  are  as  above. 

Steam  Volume  per  Stroke — Degree  of  Expansion  and  Compres- 
sion.— If  the  area  of  the  piston  is  F  square  meters,  the  volume 
of  the  entire  stroke  is  Fs  cubic  meters,  and  this  would  be  the 
volume  of  steam  used  per  stroke  if  the  steam  entered  during 
the  entire  stroke  ;  since,  however,  the  steam  enters  during  e^s, 
we  have  for  the  volume  of  steam  used 

Fe^s  cubic  meters. 

Moreover  the  piston,  when  at  the  end  of  its  stroke,  does  not 
reach  the  cylinder  cover,  but  there  is  a  space  between,  filled 
with  steam.  Also  the  steam  passages  must  each  time  be  filled. 
In  full  pressure  engines  both  these  steam  quantities  contribute 
aliaost  nothing  to  the  work.  This  space  is  hence  called  the 
"prejudicial  space."  In  expansion  engines  these  steam  quan- 
tities take  part  in  the  expansion,  and  hence  for  the  same  degree 
of  expansion  the  pressure  in  the  cylinder  sinks  less  than  when 
there  is  no  prejudicial  space,  while,  during  the  full  pressure 
period,  the  steam  in  this  space  performs  no  work.  Thus  the 
prejudicial  space  increases  indeed  the  delivery  of  expansion 
engines,  but  not  in  the  ratio  of  the  increased  consumption  of 
steam,  and  hence  an  engine  with  a  prejudicial  space  has  a  less 
efficiency  than  without. 

Let  the  prejudicial  space  be  a  fraction  e  of  the  cylinder  vol- 
ume.    Then  its  volume  is  Fes  cubic  meters. 


576  THERM0DTNAMIC8. 

If  now  steam  enters  during  tlie  entire  stroke,  tlie  steam  used 
is 

Fes  -\-  Fs  =  Fs  {1  +  e)  cubic  meters. 
In  reality,  however,  we  have 

Fes  +  Fe^s  =  Fs{e  +  e~^. 
The  steam  volume  at  the  end  of  expansion  is 

Fes  +  Fe^s  -  Fs{e  +  e^. 
Hence  the  "degree  of  expansion'"  is 

6+62 

At  the  moment  when  communication  is  closed  with  the  con- 
denser or  air,  the  inclosed  steam  volume  of  the  condenser  or  air 
pressure  is 

Fs  +  Fes  —  Fe^s  =  (1  +  e  —  eg)  Fs. 

At  this  moment  compression  begins.  At  the  end  of  compres- 
sion, the  inclosed  steam  volume  is 

Fs  +  Fes  —  FciS  =  (1  +  e  —  64)  Fs. 

Hence  the  "degree  of  compression''''  is 

_  1  +  e  -  63 
1  +  6—64 

In  the  ordinary  slide  valve,  we  have  e  about  0.05,  that  is,  the 
prejudicial  space  is  y^  of  the  entire  cylinder  volume.  In  this 
case  the  steam  passages  are  about  half  the  length  of  the  cylin- 
der. In  expansion  engines  with  two  slides,  we  have  e  =  0.07  to 
0.075.  If  the  passages  are  very  short,  as  in  the  Corliss  engine, 
where  the  prejudicial  space  is  only  that  between  piston  and 
cylinder  cover,  e  =  0.02. 


ACTION  OF  8TUAM  IN  THE  CTLINDEB.  QJ'J 


EXAMPLE. 

What  is  the  degree  of  expansion  when  the  steam  is  cut  ofE  at  half  stroke  ? 
If  we  take  e  =  0.070, 

_    0.070  +  0.5    _  0^  _  A  ^/M 
^'  ~  0.070  +  0.977  ~  1.047  ~  ""  **' 


Work  of  the  Driving  Steam. — This  work  in  every  engine  con- 
sists of  two  parts  ;  the  one  is  the  action  of  the  full  pressure 
steam,  the  other  is  that  of  the  expanding  steam.  If  ^i  is  the 
pressure  of  the  full  steam,  we  have  for  its  work 

A  =  J^sipi. 

If  we  assume  that  during  expansion,  the  steam  follows  the 
law 

we  have  for  the  work  during  expansion 


i^ib-(^ri' 


where  v  is  the  specific  steam  volume  at  the  beginning  of  ex- 
pansion. If  the  engine  uses  per  stroke  G  kilograms  of  steam, 
and  if  the  volume  of  this  weight  before  expansion  is  V  and 
after  Fi,  we  have  for  the  expansion  work 


^-^x[-(;r]- 


The  value  of  n^  varies,  as  we  have  seen,  and  depends  upon 
the  amount  of  water  in  the  steam.  On  an  average  we  can  put 
Wi  =  1.125,  when  the  steam  is  taken  directly  from  the  boiler. 
If,  however,  the  steam  is  heated,  or  even  superheated,  before 
it  enters  the  cylinder,  rhi  may  be  greater.  According  to  Gras- 
hof,  we  can  take  Wj  =  1.333,  when  the  steam  is  still  dry  after  ex- 
pansion, when,  therefore,  no  steam  condenses  during  expan- 
sion. 

37 


578  -  THEBMOBTNAMICS. 

As  to  the  value  of  2\,  it  is  to  be  taken  somewhat  less  in  find- 
ing the  expansion  work,  when  the  cut-off  is  worked  by  an 
ordinary  eccentric,  because  in  such  case  the  cut-off  is  gradual. 
"We  have,  therefore, 

Pi  =  Ai?i, 

where  /?i  is  a  proper  fraction,  which  we  can  take  about  0.95. 
In  Corliss  engines,  where  the  ports  are  suddenly  closed,  /5  =  1, 
and  pi  has  its  full  value.     In  engines  with  large  piston  velocity, 
(3  must  be  taken  much  less  than  0.95. 
The  steam  volume  V  before  expansion  is 

V=  Fes  +  Fe^s  =  Fs  {e  +  e^), 

and  the  volume  Fi  after  expansion  is 

Vi  =  Fes  +  Fe^s  =  Fs  {e  +  e^. 

If  we  insert  these  values  in  the  above  expression  for  L^,  and 
put  px  —  /?pi,  we  have 


^  ^  l3,p,Fs  (e  +  e,)  F^  _  /e  +  eA^'-'  "1 
^  %!  —  1  L        \e  +  63/         J 

or  putting  the  degree  of  expansion 


A  =  l^xPiFs  (e  +  61) 


63/ 

e  +  Ci 

e  +  €2' 

1-e 


%  —  1 
If  we  put 

/?i(e+ei)-— ^j-  =A, 

we  have 

A  =  ^iPiFs. 

The  expansion  ends  before  the  piston  arrives  at  the  end  of 
its  stroke.  Then  the  exhaust  opens,  and  the  pressure  sinks 
rapidly  to  that  of  the  condenser  or  of  the  atmosphere.  If  in 
this  case  the  mean  pressure  is  p^,  we  have,  while  the  piston 
passes  through  s  —  s., 

is  =  F{s  -  s,)pi  =  Fs(l  -  e,) p-i. 


WOMK  OF  THE  BBIVING  STEAM.  579 

Hence  tlie  total  work  of  tlie  driving  steani  is 

Xj  +  X2  +  is  =  FsiPi  +  'k'i_px  Fs  +  i^s  (1  —  63)^2 
=  e^p^Fs  +  ^xPiFs  +  Fs  il  —  e.^p^ 
=  [(ei  +  Ai)i5i  +  {l-e.:)p,-\Fs. 

WorJc  of  the  Back  Press  we. — We  must  subtract  from  the  work 
just  found  that  of  the  back  pressure.  Let  the  mean  pressure 
during  the  travel  S3  be  ps,  then 

Li  =  Fssps  =  Fse^iJ^. 

Now  compression  begins.     Let  the  law  be 

The  value  of  n^,  like  that  of  n^,  can  only  be  determined  by- 
exact  experiments.  In  the  average  n^  =  1.15  (Grashof's  Ee- 
sultate  der  mechanischen  Warmetheorie).  The  pressure  at  the 
beginning  of  compression  is  somewhat  greater  than  p^.  In 
general  we  can  take  it  /S^j^s,  where  ps  =  1.05.  If  now  we  denote 
the  volume  at  the  beginning  of  compression  by  Vo  and  at  the 
end  by  t'g,  we  have  for  the  work  of  resistance  during  com- 
pression 

But    V2  =  Fs{l  +  6  —  63)     and    Vg  =  jPs  (1  +  e  —  64),   hence 

Since  the  quantity  in  parenthesis,'the  degree  of  compression, 
is  fg, 

Lo  =  /3s2h Fs{l  +  e- 63)  '^^^  _  -^     . 
If  we  put  here 

Z5  =  ^s2hFs. 


580  THERMODYNAMICS. 

When  compression  is  ended,  the  piston  has  not  reached  the 
end  of  its  stroke,  but  has  still  to  go  s  —  s^.  While  it  goes  this 
distance,  steam  enters  from  the  boiler,  and  the  pressure  is  on 
the  average  greater  than  the  mean  pressure  of  the  driving 
steam,  esjDecially  in  high  piston  speed.     Let  it  be  pi,  then 

L,  =  F{s-Si)  23,  =  Fs{l-  €,)  i\. 
The  total  work  of  the  back  pressure  is  then 

A  +  ig  +  iye  =  Fse^p^  +  Fs\^i\  +  i^s  (1  —  64)^4 

^FsipM'r^)   +^94(1-64)]. 

If  we  subtract  this  from  the  work  of  the  driving  steam,  we 
have  the  indicated  work  per  stroke,  that  is,  without  reference 
to  friction,  working  of  pumps,  etc.     Let  this  be  Li ,  then 

L,  --=  Fsip^  (ei  +  Aj)  +  (1  -  e.)  ih-p^  (63  +  '%)  -pi{l-  64)]- 

Since  the  value  of  p,  for  ordinary  stationary  engines,  es- 
pecially Avhen  not  moving  rapidly,  is  but  little  different  from^^i, 

Li  =  Fs  [pi  (ei  +  ;\i  +  64  -  1)  +  (1  -  e^)  jp.,  -  (eg  +  ^^3)  Vz]- 

As  to  the  mean  pressure  793  from  So  to  s,  or  through  s  —  s%,  this 
depends  upon  the  pressure  at  the  end  of  expansion.  But  this 
depends  upon  the  coefficient  of  fill  ^i  of  the  cylinder.  The 
greater  this,  so  much  the  greater  is  p^,  other  things  being  the 
same.  Further,  pi  depends  upon  the  manner  in  which  the  port 
is  opened  for  discharge.  If  opened  quickly,  ^3  is  less;  if 
slowly,  p^  is  greater.     Grashof  gives 

where  ^3  is  the  mean  back  pressure.     Under  ordinary  circum- 
stances A  =  0.80. 

If  we  insert  this  value  of  ^2j  we  have 

Li  =  Fs  [j9i  (ei  + Ai  +  e4  -  1)  +  (l-e^)  {^e^Pi+ p^-'^Pz) ~Pz  (63+  ^s)]- 

If  we  divide  this  by  Fs,  we  have  for  the  mean  effective  press- 
ure 

-^  =  Pi  =  Px  (ei  +  Ai  +  64  - 1)  +  (1  -  ea)  (Aej^i  +  p^-  A^g)  -  p^ 
{e,  +  A3). 


WORK  OF  THE  BACK  PBE88UBE.  581 

If  for  brevity  we  put 

e-i_+\+e^  —1  +  le^  (1  —  e>)  —f^     and 

^3  +  ^3  —  (1  —  ^)  (1  —  ^2)  =/3j     we  have 

Hence  tlie  indicated  work  per  stroke  is 

lij)i  is  given  in  atmosplieres,  then 

A  =  10334  Fsp,  =  lOdSiFs  (pji  -  ps/s). 

Values  ofp,  2^1  and  p^. — The  pressure  2\  in  the  cylinder  can  be 
very  different  from  the  pressure  p  in  the  boiler.  This  depends 
upon  how  wide  the  valve  in  the  steam  pipe  is  open,  as  also  the 
throttle  valve  ;  also  upon  the  cross-section  and  length  of  the 
steam  passages  in  comparison  with  the  piston  speed,  which  we 
shall  denote  by  c;  upon  the  curves  and  bends  of  the  steam 
pipe ;  upon  its  length  and  radiation.  If  the  steam  pipe  is  com- 
pletely opened,  and  the  passages  have  the  required  cross-sec- 
tion, the  mean  pressure  in  the  cylinder  during  admission  varies 
but  little  from  the  boiler  pressure.  In  other  cases  the  differ- 
ence may  be  considerable.  An  exact  calculation  of  this  differ- 
ence of  pressure,  from  the  diameter  of  the  partially  opened 
steam  pipe,  the  cross-section  and  length  of  the  passages,  etc.,  is 
indeed  hardly  practicable.  It  is  evident  that  in  calculating  the 
delivery  we  must  have  regard  to  the  cylinder  pressure,  and 
when  we  wish  an  exact  determination  of  the  delivery,  we  must 
find  this  pressure  by  the  indicator.  A  long  and  narrow  steam 
pipe,  with  bends  and  angles,  and  not  protected  from  radiation, 
can  make  the  pressure  in  the  valve-box  less  than  the  boiler 
pressure.  "We  must  give  to  this  pipe  either  the  same  or  a  some- 
what less  cross-section  than  the  steam  passages. 

The  cross-section  of  these  last  depends  not  only  upon  that  of 
the  cylinder,  but  also  upon  the  mean  piston  speed  c.  Through 
these  passages  the  same  amount  of  steam  must  pass  in  a  certain 


582  THEBMODYNAMICS. 

time  as  the  cylinder  uses.  Since  tlie  velocity  of  the  steam  is 
considerable,  we  can  make  the  cross-section  of  the  passages 
less  than  that  of  the  cylinder.  But  steam,  when  it  flows  through 
relatively  narrow  passages,  experiences  a  great  loss  of  velocity, 
which  may  be  quite  considerable  when  the  valve  nearly  closes 
the  parts.  For  a  mean  piston  speed  of  1  meter,  experience 
gives  for  the  cross-section  of  the  canals  gVth  of  that  of  the  pis- 
ton. In  this  case  the  steam  pressure  in  the  cylinder  remains 
constant  during  the  entire  admission.  If  then  the  cross-section 
of  the  steam  passages  is/,  and  that  of  the  cylinder  is  F,  we  have 

We  can  further  assume  that  the  resistance,  for  the  same 
cross-section,  increases  with  the  velocity ;  that  for  2,  3,  •  •  •  c 
meters  velocity,  it  is  2,  3,  •  •  •  c  times  as  great.  In  order,  then, 
to  have  only  the  same  resistance  as  for  1  meter,  we  must  make 
the  passages  2,  3,  •  •  -  c  times  as  wide.     Hence    . 


i^"30* 

From  this  formula 

we  have  the  following  tabulation 

CROSS-SECTION   OF   STEiJI  PASSAGES. 

iston  speed  \ 

in            1=1 

1.2        1.5        2         3        4        5 

meters       ) 

/  _  1 

1           11111 

F      30       25         20       15       10      7.5      6 

Engines  with  high  j)iston  speed,  as  locomotives,  require 
therefore  wide  steam  passages.     Thus  for  c  —  2.3  meters. 

If  the  steam  passages  are  calculated  for  a  certain  piston 
speed,  and  we  let  the  engine  work  more  rapidly,  the  pressure 
in  the  cylinder  is  less  than  in  the  valve  box,  and  especially  in 


STEAM  PASSAGES.  583 

engines  working  with  little  expansion,  the  indicator  will  show 
a  noticeable  fall  of  pressure  near  the  middle  of  the  stroke,  be- 
cause here  the  velocity  of  the  piston  is  greatest. 

In  the  calculation  of  projected  engines,  we  must  consider  the 
pressure  2\  in  the  cylinder  as  given,  and  consider  the  boiler 
pressure  as  always  somewhat  greater.  In  engines  which  are 
required  to  give  great  delivery,  the  boiler  pressure  must  be 
taken  still  greater.  It  is  different  with  the  mean  pressure  ^3 
of  the  back  pressure  steam.  This  is  not  given  in  advance. 
If,  however,  the  steam  passages  are  proportioned  as  above, 
we  have  for  stationary  engines 

a,  when  non-condensing,  p^  =  1.15  atmospheres ; 
h,  when  condensing,  ^^3  =  0.2  atmospheres. 

Locomotives  which  have  a  blast  pipe,  may  have  ps  =  1.15 
to  1.27  atmospheres. 

Shorter  Form  for  the  Expressions  fi  and  %. — The  quantities  e, 
63,  ^3,  and  64  are  constant  so  long  as  the  angles  of  advance  and. 
the  laps  remain  the  same-.  Also  /?3,  iVi  and  A  are  to  be  regarded 
as  constant.  We  can,  therefore,  give  to  the  expressions  in 
which  these  quantities  occur,  a  simpler  form. 
Thus  we  have  found 

_  1  +  e  —  63 
^^"1+6-64* 

Since  now  e  =  0.05,  e-,  =  0.959,  and  e,  =  0.997, 

_  1  +  0.05  -  0.959  _  091  _  .,    -. 
^'~  1  +  0.05  -  0.997  ~  0.53  ~ 

Also  /?3  =  1.05,  and  n^  —  1.15,  hence 

A3  ==  y?3 (1  +  e -  63)  ''"'"' T"*"  =  0.0538. 

Since  A  can  be  taken  0.8, 

•3  =  63  + A3- (1- A)  (1- 62) 
=  0.959  +  0.0538  -  (1  -  0.8)  (1  -  0.977)  =  1.0082. 


584  THEBM0DTNAMIC8 

Finally, 
/j  =  ei  +  A, +  64-1  +  ^61(1-62)  =  ei  +  Ai  + 0.997 -l  +  0.8ei  (1-0.977) 
=  1.0184ei  +  Aj  -  0.003. 

Shortest  Form  of  the  Formula  for  the  Indicated  Delivery. — The 
formula  already  found  for  the  effective  pressure  ]Ji  becomes 
now 

^=p.=  p^  (1.0184:6,  +  Ai  -  0.003)  -psx  1.0082, 

or 

X.  =  Fs2J^  =  Fs  ip,  (1.0184ei  +  A^  -  0.003)  -  1.0082^^3], 

where  pi  and  p^,  are  the  pressures  in  kilograms.    If  p^  and  p^  are 
given  in  atmosj)heres, 

Li  =  10334i^s  [pi  (1.0184ei  +  A,  —  0.003)  -  1.0082^3]- 

So  soon  then  as  we  know  for  any  engine  (which  must  have 
the  assumed  angle  of  advance,  etc.),  the  coefficient  of  fill  61  and 
K  we  can  find  A-.     Also,/i  =  1.0184ei  +  1^-  0.003. 

Grashof,  in  his  "Eesultaten  der  mechanischen  Warmetheorie," 
has  found  ft,  '^•L,f\  for  different  values  of  e,,  as  given  by  the 
table  below.  The  table  also  contains  A^i"'j  which  gives  the 
pressure  Ai^i^i'*'  at  the  end  of  expansion. 


^1 

^i 

A, 

/i 

/i.^i"' 

0.1 

0.1623 

0.2627 

0.3615 

0.128 

D.15 

0.2100 

0.2963 

0.4461 

0.164 

0.3 

0.2578 

0.3199 

0.5206  . 

0.207 

0.25 

0.3055 

0.3351 

0.5867 

0.350 

0.3 

0.3533 

0.3428 

0.6453 

0.295 

0.35 

0.4010 

0.3444 

0.6978 

0.840 

0.4 

0.4488 

0.3404 

0.7448 

0.386 

0.45 

0.4965 

0.3312 

0.7865 

0.433 

0.5 

0.5448 

0.3171 

0.8233 

0.479 

0.55 

0.5920 

0.2987 

0.8558 

0.537 

0.6 

0.6398 

0.2765 

0.8845 

0.575 

0.65 

0.6875 

0.2501 

0.9091 

0.633 

0.7 

0.7353 

0.2206 

0.9305 

0.673 

0.75 

0.7830 

0.1876 

0.9484 

0.731 

0.8 

0.8308 

0.1514 

0.9631 

0.771 

0.85 

0.8785 

0.1119 

0.9745 

0.821 

0.9 

0.9263 

0.0700 

0.9836 

0.873 

0.91 

0.9358 

0.0618 

0.9855 

0.883 

INDICATED  DELIVERY.  585 

From  these  values  the  values  of  pi  in  the  following  tables 
are  calculated  for  different  values  of  Cx  and  pi.  The  first  table 
applies  to  non-condensing  engines,  the  other  to  condensing. 
In  the  first,  the  mean  back  pressure  p^  is  1.1 ;  in  the  other,  0.2 
atmospheres.     Cases  in  which  A2^ifi"'  <  Ih  ^^^  excluded. 

Pi  FOE  NON-CONDENSING  ENGINES 


i'i=3 

^1  =  4 

i^.=5 

i'i=6 

i?.=7 

i>i  =  8 

i',=9 

r=0.1 

3.145 

0.15 

3.014 

2.460 

2.906 

0.2 

2.015 

3.535 

3.056 

3.577 

0.25 

1.825 

3.411 

3.998 

3.585 

4.171 

0.3 

1.472 

2.118 

3.763 

3.408 

4.054 

4.699 

0.35 

1.682 

2.380 

3.078 

3.776 

4.474 

5.171 

0.4 

1.126 

1.870 

2.615 

3.360 

4.105 

4.850 

5.594 

0.45 

1.251 

2.037 

2.824 

3.610 

4.397 

5.183 

5.970 

0.5 

1.361 

2.184 

3.008 

3.831 

4.654 

5.478 

6.301 

0.55 

1.459 

2.314 

3.170 

4.026 

4.883 

5.788 

6.593 

0.6 

1.545 

2.429 

3.314 

4.108 

5.083 

5.967 

6.853 

0.65 

1.618 

2.528 

3.437 

4.346 

5.255 

6.164 

7.073 

0.7 

1.683 

2.613 

3.544 

4.474 

6.405 

6.335 

7.266 

0.75 

1.736 

2.685 

3.633 

4.583 

5.530 

6.478 

7.427 

0.8 

1.780 

2.744 

3.707 

4.670 

5.633 

6.596 

7.559 

0.85 

1.815 

2.789 

3.764 

4.738 

5.713 

6.687 

7.663 

0.9 

1.842 

2.826 

3.809 

4.793 

5.776 

6.760 

7.744 

0.91 

1.848 

2.833 

3.819 

4.804 

5.790 

6.775 

7.761 

pi    FOR  CONDENSING   ENGINES. 

p,  =1.5 

i),  =3 

p,  =2.5 

P,  =3 

i^,=4 

P.  =5 

i>,  =  6 

=  0.1 

0.521 

0.702 

0.883 

1.244 

1.606 

1.967 

0.15 

0.468 

0.691 

0.914 

1.137 

1.583 

2.029 

3.475 

0.3 

0.579 

0.840 

1.100 

1.360 

1.881 

2.401 

3.922 

0.25 

0.678 

0.972 

1.265 

1.558 

3.145 

2.733 

3.319 

0.3 

0.766 

1.089 

1.412 

1.734 

3.380 

3.025 

3.670 

0.35 

0.845 

1.194 

1.543 

1.893 

3.590 

3.287 

3.985 

0.4 

0.916 

1.288 

1.660 

3.033 

3.778 

3.523 

4.267 

0.45 

0.978 

1.371 

1.765 

3.158 

2.944 

3.731 

4.517 

0.5 

1.033 

1.445 

1.857 

3.268 

3.092 

3.915 

4.738 

0.55 

•     1.082 

1.510 

1.938 

2.366 

3.222 

4.077 

4.933 

0.6 

1.125 

1.567 

3.010 

2.453 

3.336 

4.331 

5.105 

0.65 

1.162 

1.617 

2.071 

3.526 

3.435 

4.344 

5.353 

0.7 

1.194 

1.659 

2.125 

2.590 

3.530 

4.451 

5.381 

0.75 

1.221 

1.695 

2.169 

2.644 

3.593 

4.540 

5.489 

0.8 

1.243 

1.725 

2.206 

2.688 

3.651 

4.614 

5.577 

0.85 

1.260 

1.747 

2.235 

2.722 

3.696 

4.671 

5.645 

0.9 

1.274 

1.766 

2.357 

2.749 

3.733 

4.716 

4.700 

0.91 

1.277 

1.769 

3.363 

2.755 

3.740 

4.736 

5.711 

586  THEBMOD  YNAMIGS. 

From  these  two  tables  we  can  take  the  mean  effective  press- 
ure for  different  coefficients  of  fill,  and  then  can  find  the  indi- 
cated work  per  stroke  from  the  formula, 

If  u  is  the  number  of  revolutions  per  minute,  and  hence  2u 
the  number  of  strokes,  the  indicated  work  per  second  is 

2mX^  _  10334  X  "^upiFs 
60    ~  60 


~~  =  10334:^3^  —  Fs  meter-kilograms. 

In  horse  powers 

^j  uL,  10334  ^ 

^^=  30^75  =  3(nr75^'^^^^- 

Since  the  engine  makes  in  1  minute  2u  strokes,  and  each 
stroke  is  s  meters  long,  the  distance  passed  over  by  the  piston 
per  minute  is  ^iis  meters,  and  in  one  second 

2ms  _  lis 
"60""  30 

This  is  therefore  the  mean  velocity  c  of  the  piston. 
We  have  then  c  in  place  of  7^ ,    and 

,^       10334     p 


Example  1. — The  diameter  of  the  cylinder  of  a  non-condensing  engine  is  0.47 
meters,  the  stroke  s  —  1.044  meters,  the  number  of  revolutions  per  minute  is  24, 
and  the  pressure  in  the  cylinder^,  =  4  atmospheres.  What  is  the  delivery  per 
second  when  the  proportions  of  the  slide  valve  are  as  has  been  assumed  in  our 
discussion  ? 

First,  F=  ^  =  0.7854d«  =  0.7854  x  (0.47)^  =  0.1734. 


THEORY  OF  THE  CRANK 


587 


Now  from  our  tables,  for  n  pressure  of  4  atmospheres  and  e^  —  0.91,  pt  = 
2.833.     Hence 


10334  X  3.^ 


0.1734  X  1.044  =  5301  m.  kil. 


and 


Ni  =  4.593  X  24  X  3.833  x  0.173  x  1.044  =  56.4  horse  power. 


Example  3.— What  wouLT  be  the  delivery,  if  Ci  =  0.4  ? 
In  this  ease 

Pi  =  1.870,     and 

Ni  =  4.593  X  24  X  1.87  x  0.173  x  1.044  =  37.33  horse  power. 

* ' 

37  33 
The  delivery  is  then  —^  ,  or  0.66  of  the  first,  while  only  about  0.4  as  much 

steam  is  used,  and  hence  not  half  as  much  fuel. 

Work  of  the  Engine  token  Disconnected. — From  the  indicated 
work  of  the  engine,  just  calculated,  we  must  subtract  that  re- 
quired to  work  the  pumps,  eccentrics,  and  overcome  the  va- 
rious frictional  resistances.  We  obtain  this  work  if  we  deter- 
mine that  of  the  disconnected  engine,  and  then  increase  this 
latter  by  a  certain  amount,  as  given  by  experiment ;  because 
the  work  required  by  the  friction  of  a  ivorking  engine  is  greater 
by  a  certain  amount  than  that  of  an  engine  running  without 
overcoming  useful  resistance. 

The  greatest  part  of  the  work  in  question  is  required  to  run 
the  fly-wheel.     Let  us  estimate  it  first. 

Theory  of  the  Crank.- — The  fly-wheel  serves  not  only  to  con- 
vert reciprocating  motion  into  circular  or  rotary,  and  to  carry 


P 


588 


THEBMOD  TNAMIC8. 


tlie  motion  past  the  dead  points  U  and  0,  but  also  to  make  tlie 
motion  uniform. 

If  BA,  Fig.  102,  is  the  connecting  rod,  CA  the  crank,  the  rod 
exerts  either  compression  or  tension  upon  the  crank.  If  the 
rod  has  the  position,  and  rotation  takes  place  as  shown  by  the 
arrows,  it  acts  to  cause  compression.  If  the  crank  is  at  ^i,  or 
A^,  the  rod  causes  tension.  Let  AR  =  (J  be  the  pressure. 
This  pressure  only  acts  in  part  to  cause  rotation.  That  part, 
namely,  which  is  perpendicular  to  the  direction  of  the  crank, 
or  which  acts  in  the  direction  AX.  By  the  other  component  of 
Q,  the  crank  is  forced  against  the  bearings  and  the  friction  in- 
creased. If  now  we  decompose  AR  =  Q  into  the  directions 
AX  and  A  G,  AD  is  the  force  causing  rotation,  and  AE  is  that 
causing  compression.     If  the  angle  DRA  —  RA  G  =  y,  we  have 


and 


AD  =  Q  sin  y. 
AE  =  Q  cos  y. 


(1). 


The  resistance  opposes  the  force  Q  sin  y.   Let  it  be  AP  =  P, 

and  constant  at  all 
points  of  the  crank 
circle.  We  assume 
that  the  work  of  rota- 
tion is  equal  to  that  of 
the  resistance.  There 
are  then  four  points 
in  which  the  driving 
force  Q  sin  y  is  equal 
to  the  resistance  P. 
These  points  can  be 
easily  found.  For  this  purpose  we  assume,  for  the  sake  of 
simplicity,  that  the  connecting  rod  is  very  long  compared  to 
the  crank.  The  results  thus  obtained  deviate,  as  we  shall  see 
later,  but  little  from  those  obtained  under  the  assumption  of  a 
finite  rod.  If,  therefore,  the  rod  is  assumed  very  long,  it  is 
always  parallel  to  UO,  and  we  have 

Qsiny  =  Q  sin  AGU  =  P    .    .     .     .     (2). 
and 

sin  /  =  sin  AGU  ~  jj  -     •     •     •     •     (3)- 


THEORY  OF  THE  CRANK  589 

Let  us  assume  the  engine  to  work  with  full  pressure.  The 
force  Q  —  AJR  remains  then  constant  during  the  whole  dis- 
tance UO  =  2r.  The  work  of  this  force  is,  for  an  entire 
revolution,  4:rQ. 

The  work  of  the  resistance  F  in  the  same  time  is  2^rP,  and 
since  both  works  are  equal, 

2Q=  TzP (4). 

or 

F  =  —  Q  =  0.6366Q  ....         (5). 

The  resistance  cannot  be  greater  for  uniform  motion.  If  we 
insert  this  value  in  (1)  we  have 

sin;/  =  sin  ^(7  Z7  =  0.6366, 
or  the  angle 

r  =  Acu^3rs2'. 

When,  then,  the  crank  makes  the  angle  39°  32'  with  UO,  the 
moving  or  tangential  force  is  equal  to  the  resistance  F.  For 
every  other  position  of  the  crank  the  tangential  moving  force 
varies  with  the  angle.  If  we  make  A^CO,  A^CO,  ^gC?/ equal  to 
ACU,  we  have  at  the  points  A^,  A^,  ^3,  the  moving  force  Q  sin 
y  equal  to  the  resistance  P.  Between  A  and  ^1,  and  between 
A2  and  A^  the  moving  force  is  greater,  while  between  A^  and 
A^,  and  A^  and  A,  it  is  less  than  the  resistance.  But  since  the 
work  of  the  force  must  be  equal  to  that  of  the  resistance,  the 
excess,  in  the  first  portions,  must  equal  the  deficiencies  in  the 
other  two.  In  order  to  accomplish  this,  it  is  necessary  to  at- 
tach a  heavyweight  to  the  axle.  Such  a  weight  is  the  fly  wheel. 
The  object  of  the  fly  wheel,  then,  is  to  receive  the  excess  of 
work  on  AAi  and  ^2^3?  and  give  it  up  along  A1A2,  and  A^A. 

f  Galeulation  of  the  Weight  of  Fly  Wheel. — We  can  now  easily 
find  the  weight  8  of  the  fly  wheel.  Suppose,  first,  a  weight 
81  upon  the  crank  pin,  which  has  the  same  living  force 
as  the  fly  wheel,  or,  more  simply,  of  the  fly  wheel  rim.  If  the 
mean  velocity  of  the  weight  is  v  and  that  of  rim  F,  we  must 
have,  when  the  one  mass  is  replaced  by  the  other,  S^v^  =  SV-, 
or 

8  =  8AV) (6). 


<t) 


590  THEBMODYNAMIGS. 

If  the  mean  radius  of  the  rim  is  i?,  and  of  the  crank  r  =  -^, 

A 

we  have 

v:V::r:R,     or     (f /  =  (^   •     •     (7). 
Hence  from  (6) 

«=«'(iy («)■ 

We  have  now  to  determine  the  weight  8-^. 
Since  from  ^i  to  A^  there  is  an  excess  of  force,  there  is  an 
increase  of  velocity.     If  the  velocity  at  A  is  v^  and  at  A^,  V2, 

the  livinsj  force  Sii  Ais  ^  Su  and  at  Ai,  7^  S^. 

Hence  from  A  to  Ai  the  living  force  stored  is 

^^. («)• 

This  work  is  given  up  from  A^  to  A2,  absorbed  again  from  A^ 
to  As,  and  so  on.     If  now  the  mean  velocity  is  v,  then 

h_±3:.  =  ^,,     or     V,  +  V2  =  ^v.      ...     (10). 

We  denote  the  ratio  of  the  difference  between  the  greatest 
and  least  velocities  {v^  —  Vi),  to  the  mean  velocity,  by  the  term 
*'  coefficient  of  irregvlariti/,"  and  represent  it  by  d.     Thus 

6  =^^~^'   ,     or    i)d  =  v,-v^.     .     .     (11). 

This  coefficient  must  be  taken  less,  according  as  more  uni- 
formity is  required.  Thus  the  coefficient  should  be  much  less 
for  an  engine  required  to  run  a  cotton  mill,  for  instance,  than 
for  working  pumps,  etc. 

From  (10)  and  (11) 

v^  -  v^  =  2i7M. 

Inserting  this  in  (9)  we  have  for  the  increase  of  living  force 
from  A  to  Ai, 

^«. (12). 


WEiaHT  OF  FLY  WHEEL.  591 

We  can  now  obtain  another  expression  for  tliis  work.     The 
force  Q  performs  from  A  to  A^  the  work 

^Qr  cos  y  =  2Qr  cos  39°  32'  =  lM2iQr. 

The  resistance  P  =  0.6366^  is    overcome    through  ^Jli,  or 

through  an  arc  of  180  -  2  (39"  39)  =  100°  56'.     The  length  of 

,,  .  .    100°  56'  T_-,^ 

this  arc  is  — zt-^t, —  nv  =  1.7615r. 
ioO 

Hence  the  work  of  the  resistance  is 

L7615r  X  0.6366Q  =  lA2UQr. 

The  excess  of  the  work  of  the  force  is  then 

(1.5424  -  1.1214)  Qr  =  0.4210  ^r. 

This  excess  must  be  equal  to  that  of  the  living  force. 
Hence 

—  S,  =  0.4210  Qr,     or    v'S^  =  0.4210  ^  g. 


Since        S^i^  =  SV' =  8  (  y  v)'=  S  (y)'^'.  ^e  have 


S  (—yv^  =  0A210^g,   or 


'-'■^'^'(iy%^ (1^)- 


For  a  finite  connecting  rod  v^Si  is  somewhat  greater.     If,  for 
example,  the  connecting  rod  is  6  times  as  long  as  the  crank,  or 


-=  1-,  we  find  by  similar  calculation 


^  =  0.4978  f-^)'^^,. 


592  THEBMODTNAMICa. 


For 

-i-^ 

^=«-5^^<^)^4^- 

For 

;=* 

S  =  0.6436  (01^  <,. 

If  tlie  engine  works  expansively,  then  for  tlie  same  delivery 
and  otherwise  similar  circumstances,  tlie  above  expressions 
must  be  multiplied  by  tlie  expression 

0.77  +  0.23  i  -  0.017  (^iV   ....     (14). 

where  ei  is  the  coefficient  of  fill  of  the  cylinder.     Thus  for  e^  = 
0.5,  or  for  cut-off  at  |  the  length  of  cylinder, 

0.77  +  0.23  X  2  -  0.017  x  4  =  1.162. 

We  shall  now  express  Qr  in  terms  of  the  indicated  work  L^. 
The  force  Q  represents  the  mean  effective  pressure  upon  the 
piston,  hence 

where  pi  is  given  in  kilograms  per  sq.  meter.     Hence 

Qr  =^  rFp,  =  \sFp,  =  J-Z, 

Substituting  this  in  the  above  expressions,  and  inserting  the 
value  of  g  ~  9.81  meters,  we  have  for 

r  _  ^ 
I  "' 

r  \2  L 


r  _  ^ 


^=2-^39 '^;^ 


S  =  2.525  (p"-''' 


WEIGHT  OF  FLY  WHEEL.  593 


The  weight  B  is  that  which  when  applied  at  the  distance  B 
from  the  center  G  of  the  shaft,  will  cause  the  required  degree 
of  uniformity  of  motion. 

Now  every  fly  wheel  consists  of  a  rim  and  4  or  8  arms.  If 
we  let  8  refer  to  the  rim  alone,  then  by  reason  of  the  inertia  of 
the  arms  the  degree  of  irregularity  will  be  less  than  that  as- 
sumed in  the  calculation.  We  may,  therefore,  take  about  0.9  of 
the  above  values  for  8  and  consider  this  as  the  weight  of  the 
rim.     We  have  then 


(1.)  for    [  -  i 

^.  =  2.195  (^)j^;, 

(2.)  for    [=.i 

^-^•^^K^y^' 

(3.)  for    1=1 

«.  =  2.398  (^yi^. 

These  formulae  apply  to  full  pressure  engines.  For  engines 
working  expansively,  we  must  multiply  by  the  coefficient  given 
by  (14). 

If  the  arms  are  fVtlis  of  the  weight  of  the  rim,  and  if  we  de- 
note the  weight  of  the  fly  wheel  shaft  by  8^^,  and  the  total 
weight  of  shaft  and  wheel  by  8,  we  have 

^S'^  810  + 1.3^,.. 

EXAMPLE. 

What  weight  must  the  fly  wheel  have  for  the  engine  of  page  586,  of  56.4 
horse  power,  or  Li  --=  5301  meter-kilograms  ?    The  coefficient  of  irregularity  8  is 

taken  ./^j-,  and  -y  =  ^  and  -^  =  \, 
LI  jU 

38 


594  THERMODYNAMICS. 

The  circumference  of  the  crank  circle 
lutions  per  minute,  the  mean  velocity  v 


The  circumference  of  the  crank  circle  is  2nr  =  its.     Since  there  are  u  revo- 

nv.s        3.1416  X  1.044  x  24 


CO  60 

meters.     Hence 

8r  =  3.273  (4)2  ,,  oo!?^  -r  =  §862  kilograms. 


If  with  the  same  delivery  tlie  coefficient  of  fill  is  e^  =  0.5,  we 
must  multiply  this  weight  by  0.77  +  0.23  x  2  -  0.017  x  4  = 
1.62,  and  hence 

Sr  =  10298  kilograms. 

The  weight  of  the  entire  fly  wheel,  in  the  first  case,  is  1.3  x 
8862  =  11521,  and  in  the  second  1.3  x  10298  =  13387  kilograms. 

Since  the  friction  of  the  fly  wheel  journals  consumes  a  con- 
siderable amount  of  the  work,  we  should  have  the  weight  as 
small  as  possible,  and  make  the  radius  as  great  as  possible. 

Dimensions  of  the  Bim  and  Arms. — If  we  know  the  weight  of 
the  rim,  we  can  easily  find  its  dimensions.  We  denote  the 
cross-section  by  F^  and  the  weight  of  1  cubic  meter  of  cast  iron 
by  ;/.  (Since  the  specific  weight  of  cast  iron  is  7.4,  1  cubic  me- 
ter weighs  1000  x  7.4  =  7400  kilograms.)  The  volume  of  the 
rim  is  then 

^TtRFi  cubic  meters, 
and  its  weight 

"We  have  then 


IrtRF^Y- 

Sr  =  27r  RF,r, 

or  putting  y,  7400  kilograms, 

Sr  =  4:64.72RF,, 
and  hence 

Sr 


F,= 


46472i2 


In  general,  we  make  the  radial  depth  of  rim  1  to  2  times 
the    thickness.     If  the  first   is   d  and  the  second  h,  we  have 

Fi    =r    bd: 


nniENSIONS  OF  RIM  AND  ASMS.  595 

If  we  denote  tlie  number  of  arms  by  n,  and  the  cross-section 
by  F2,  tlie  weight  is 

74:00nF2B, 
and  we  have 

iSr  =  liOOjiF^Ji. 

To  smaller  wheels  we  may  give  4  arms,  to  larger  6  or  even  8. 
If  n  is  given,  we  have 

Sr 


F.= 


22200«i? 


EXAMPLE. 

What  must  be  the  cross-sections  F^  and  F.^  for  the  fly  wheel  of  the  full 
pressure  engine,  already  mentioned,  for  which  the  weight  of  fly  wheel  rim  has 
been  found  8862  kilograms,  and  the  mean  radius  i?  =  2.61  meters  ? 

8862 
We  have  F^  =    ,.  ^^ — -  ^  „^    —  0.0729  square  meters.     If  we  take  b  =  ^.d,  we 
^      46472  X  2.61  '■ 

have 

Fi  =  0.0729  =^(^2,  ord=  i  0.1094  =  0.331  meters,  or  33.3 

centimeters,  and  h  =  %  x  33.3  =  22.2  centimeters. 

For  the  arms,  if  we  have  8  of  them. 


=  0.015  sq.  meters. 


22200  X  8  X  2.61 


Diameter  of  the  Journals —  Weight  of  the  Shaft. — We  can  now 
find  the  weight  of  the  shaft.  The  journals,  according  to  Morin, 
should  have  the  diameter 

di=  20  i/  —  centimeter 


f^ 


where  iV^  is  the  indicated  horse  poM^er  and  u  the  number  of 
revolutions  per  minute.  The  diameter  of  the  shaft  can  be  1  or 
2  centimeters  greater. 

For  our  full  pressure  engine,  Ni  =  56.4  horse  power,  w  =  24; 
hence 

c^i  =  20 1/^  =  26.62  centimeters. 


596  THERMODYNAMICS. 

If  we  make  the  diameter  of  tlie  shaft  28.62  centimeters,  and 
assume  it  is  3  meters  long,  the  weight  is  approximately 

7400  X  3.14  (0.143j-  x  3  =  1421  kilograms. 

Hence  the  weight  of  fly  wheel  and  shaft  is    . 

S  =  1421  +  11521  =  12942  kilograms. 

Mean  Effective  Pressure  Necessary  for  Overcoming  the  Resist- 
ance of  Friction. — Since  we  now  know  the  diameter  d^  of  the 
shaft  and  journals,  and  the  weight  of  fly  wheel  and  shaft,  we 
can  find  the  work  required  to  overcome  the  journal  friction. 

The  circumference  of  the  journal  is  Ttd^.  If  we  put  coefficient 
of  friction  =  Ci,  we  have  for  the  work  per  stroke  (per  half  revo- 
lution) 


If  ^,s  is  the  mean  effective  pressure  per  square  meter  of  the 
piston,  required  for  this  work,  the  work  of  the  steam  per 
stroke  is 

10334p,i^s  =  10334  -^^  %  , 

where  d  is  the  diameter  of  the  piston.     We  have  thus  }%, 

_  4  X  l-rrdiCiS  _      MiCiS 
^'~  T0334^d¥"  ~  10334^" 

If  we  take  the  coefficient  of  friction  Ci  =  0.1,  we  have 

«,-:  0.00002^^. 

^  d-s 

For  the  mean  effective  pressure  pi,  required  to  overcome  the 
friction  of  the  piston,  piston-rod,  cross-head,  crank-pin,  eccen- 
tric, slide  valve,  and  feed  pump,  Grashof  gives 

0.0227 

Pt  =--    —3—- 


MEAN  EFFECTIVE  PRESSURE  FOR  FRICTION.  597 

We  have  accordingly  for  tlie  mean  effective  pressure  of  the 
engine  when  disconnected, 

p,„^  0.00002-^^  +  ^. 

For  condensing  engines,  we  have  also  the  air  and  cold  water 
pumps. 

Mean  Effective  Pressure  required  for  tuorldng  the  Cold  Water 
and  Air  Pumps.— li  we  assume  that  the  engine  requires  per 
hour  D  kilograms  of  steam,  and  n  times  as  much  cold  Avater  for 
condensing  this  steam,  the  cold  water  pump  must  furnish  per 
hour  nB  kilograms  of  cold  water.  If  the  height  is  h  meters, 
the  work  required  to  furnish  this  water  is  per  hour  nDh  meter- 
kilograms. 

We  may  allow  that  at  each  stroke  xV^^i  o^  the  water  falls 
back,  that  therefore  not  oil),  but  (1  +  0.1)  7iD  —  l.lnD  kilo- 
grams must  be  raised  the  distance  Ji.  Hence  the  work  per 
hour  is 

l.lnDJi. 

If  we  allow  ^d  of  this  for  resistance  of  friction,  we  have  for 
the  actual  work  required  per  hour, 

f   X   ^7lDk, 

and  per  minute,  about 

l.SnWi 


60 

If  now  2^g  is  the  mean  effective  pressure  upon  the  piston  re- 
quired to  perform  this  work,  we  have  for  the  vv'ork  of  the  steam 
per  minute  (;u  revolutions), 

10334  X  2uMpg. 

This  work  must  equal  the  preceding,  hence 

1.5nDJi 


10334  X  2uFsp^ 


60 


0.00015  :,„^-^-    very  nearly. 


598  THERMODYNAMICS. 

Let  us  now  determine  the  work  required  by  the  air  pump. 
This  has  to  remove  the  water  weight 

{n+l)D 

kilograms  from  the  condenser.  This  requires  the  space 
0.001  [n  +  1)  D  cubic  meters.  In  removing  the  water  from  the 
condenser,  the  pressure  overcome  is  that  of  the  atmosphere  less 
the  condenser  pressure.  Upon  the  return  stroke,  the  con- 
denser pressure  is  overcome.  In  each  double  stroke  the  aver- 
age pressure  is  then  that  of  the  atmosphere.  The  work  is 
then 

10334  X  0.001  {n  +  1)D  meter-kilograms  per  hour, 

or 

—nor-   X  0.001  {n  +  1)D  per  minute. 

lipe  is  the  mean  effective  jDressure  which  performs  this  work, 
we  have 

10334|3,i^s2w  =  ^~  X  0.001  {n  +  1)D, 


^'='>-^'''^,~- 


Since  the  pump  must  also  move  the  air  from  the  condenser, 
and  the  frictional  resistance  must  be  overcome,  we  should,  ac- 
cording to  Grashof,  at  least  double  this  pressure,  and  thus 
have 

p^  =z  0.C022  \  ^^  „^ —  . 


If  we  neglect  1  in  comparison  to  n, 

Pe  =  0.0022  ^^ 


USEFUL  DELIVERY.  599 

"We  have,  then,  for  the  mean  effective  pressure  in  the  cylin- 
der, required  for  working  the  cold  water  and  air  pumps, 

P,+Pe=  P'r.  =  0.00015  ,-^,  +  0.0022      ''^ 


120i^5M  120i^5W 

As  a  rule,  n  —  20,  and 

D 


p\n  =  (0.003A  +  0.045) 


120i^sw' 


Hence  the  mean  effective  pressure  p^n  of  the  engine,  when 
disconnected,  if  condensing,  is 

AAAAAo  d^S       0.0227 
p„i  =  0.00002  -j^  +  — ^—  +i>m, 


0.00002  1^+^^ 
D 


+  (0.003A  + 0.045)-,^ 

Useful  Delivery. — The  work,  therefore,  of  the  engine  when 
disconnected,  per  stroke  is  L^  =  Fsp^,  where  ]3m  is  to  be  found 
as  above  for  condensing  or  non-condensing  engines.  If  we  sub- 
tract this  work  from  the  indicated  delivery,  we  have  the  useful 
work  per  stroke.     For  this,  then, 

Lu  —  Li  —  Lm- 

Now  it  is  evident  that  the  work  L^,  absorbed  by  the  different 
frictional  resistances  and  by  the  working  of  the  pumps,  is 
greater  when  the  engine  is  at  work  than  when  disconnected. 
According  to  Pambour,  we  must  increase  the  work  of  the  re- 
sistances by  a  part  of  the  useful  work,  and  then  subtract  from 
Li.     Thus  he  takes  this  part  at  0.12  to  0.14i^«,  and  hence 

A,  =  Li-  {L„,  +  O.ldL,), 


600  THERM0DTNAMIC8. 

or 

Lu  +  0.13Z,  =  Li  -  A^, 
or 

J    _  Lj  —  i,,^ 

^"  ~   i:i3   • 

Hence,  iip^i  is  the  mean  effective  pressure  required  for  the 
useful  work, 

_  Fsp,  ~  Fsp^ 
J^sp^  -  133         ' 


_Pi-Pra 

Pu-     ^-^3    . 

If  we  have  u  revolutions  j)er  minute,  the  horse  power  of  use- 
ful work  is 

^      uL^      _  Fcp,, 
"      30  X  75  ~     75     ' 

where  p^  is  in  kilograms  per  square  meter. 

For  the  indicated  efficiency  {m)  of  the  engine,  we  have 

Li       N,      p, 

Steam  Weight  per  Hour. — According  to  Yolkers,  the  steam 
weight  per  hour  D  is  given  by 

D  =  nOFsu  [(e  +  ei)  y^  -  ey,']  +  4:50d  V^  • 

Here,  y-i  denotes  the  weight  in  kilograms  of  1  cubic  meter  of 
steam  at  the  pressure  jJi  atmospheres,  which,  for  saturated 
steam  is  given  by  Table  II. 

For  non-condensing  engines,  j^o  =  1.32 ;  and  for  condensing, 
0.264 

The  second  term  includes  the  loss  of  steam  and  heat,  as  also 
the  moisture  of  the  steam,  for  engine  in  average  good  con- 
dition. 

Finding  thus  B,  we  can  find  j^9,„  from  the  formula  already 
given  for  that  quantity. 


DIMENSIONS   OF  CONDENSER   AND   PUMPS.  601 

Dimensions  of  the  Condenser  and  Pumps. — "We  have  seen,  page 
469,  that  the  ratio  n  of  the  weight  of  cooling  water  and  con- 
densed steam  is  given  by 

&m-t 

^  =  ^7^4' 

where  t^  is  the  temperature  of  the  condenser  water,  and  ^o  that 
of  the  injected  cokl  water. 

If  we  take  as  a  mean  temperature  t^  =  46°,  corresponding  to 
a  pressure  of  y^th  atmosphere,  and  t^  =  18°,  we  have 

600  -  46       554 

If  ^0  is  less  than  18°,  n  may  be  less  than  20. 
The  volume  of  the  condenser  is  taken 

n     Fs  ,    Fs 
C=-^to-3-, 

that  is,  ^d  or  |th  of  the  cylinder  volume. 

Let  now  Fi  be  the  volume  described  by  the  piston  of  the 
cold  water  pump  per  stroke,  then 

2m  Fi 

is  the  space  described  per  minute.  If  the  pump  is  double  act- 
ing, this  is  the  water  quantity  furnished  per  minute.  If  it  is 
single  acting,  the  quantity  is  uV-i  cubic  meters. 

In  the  first  case,  the  water  per  hour  is  2  x  Q^uVi,  and  in  the 
second  QOuV-^  cubic  meters. 

Now  the  steam  weight  per  hour  is  D,  and  the  water  weight 
required  for  condensation  is  nD.     The  entire  weight  {n  +  1)  -D 

takes  the  space  ^       ^ —  cubic  meters. 

We  have  then  for  V]_ 

(a.)  For  a  donhle  acting  suction  pump, 


g02  THERMODTNAMIGS. 

If  about  10  per  cent,  of  water  falls  back,  we  must  increase  Vi 
by  Jotli.     Tlien 

{n  +  1)D  ^  (n_+  IJIl 
'  ~~        120000m        109091W  ' 


for  wbicli  we  may  put 


^^  =  wmh.'"'^''"^''^''- 


(h)  For  single  acting  pump, 
we  bave  double  tbis,  or 


^'  =  -5^7™^'"™*'''^'- 


Quantity  of  Fuzl  (B)  j^er  Hoicr. — Wben  we  know  tbe  steam 
weigbt  I)  per  bour,  we  can  easily  find  tbe  amount  of  fuel  re- 
quired for  tbe  generation  of  tliis  steam,  wben  tbe  beating  value 
of  tbe  fuel  and  tbe  efficiency  of  tbe  boiler  and  grate  are  known. 

We  know  tbat  1  kilogram  of  steam  requires  for  its  generation 

IF=  606.5  +  0.305^  beat  units. 

I)  kilograms  tben  require  WD. 

If  1  kilogram  of  fuel  furnisbes  by  complete  combustion  K 
beat  units,  we  bave  for  JVI) 

-^—  kilograms  of  fuel  necessary, 

wben  all  tbe  beat  goes  to  generate  steam. 

But  only  a  part  of  tbe  fuel  is  completely  consumed,  even  if 
tbe  greatest  part.  Tben,  tbe  bot  gases  of  combustion  carrj'- 
off  a  large  amount  of  beat,  and  only  a  part  of  K  is  effective 
to  beat  tbe  boiler  plates.  Tbis  part,  tbe  ratio  of  wbicb  to 
tbe  total  beating  power  B  of  tbe  fuel  we  call  tbe  efficiency  of 
tbe  grate,  we  denote  by  Wi.  Furtber,  not  all  tbe  beat  wbicb 
enters  tbe  boiler  plates  goes  to  beat  tbe  water.  A  part  is  lost 
by  reason  of  radiation  and  imperfect  conduction.  Tbe  beat 
wbicb  tbe  water  actually  receives,  compared  to  tbat  received 


QUANTITY  OF  FUEL  PER    HOUB.  603 

by  tlie  boiler,  is  the  efficiency  of  tlie  boiler.     We  denote  it  by 

It  is  evident,  then,  that  for  vaporizing  D  kilograms  of  water 
"we  must  have 

WD 

B  =  rrr  ttt  ^  kllograms  of  fuel. 
Wx  JV^K        ^ 

Cost  of  a  Horse  Power  per  Hour. — Let  us  now  determine  the 
cost  of  a  horse  power  per  hour.  Let  the  price  of  the  boiler  be 
P  and  of  the  engine  P^.  Also  the  price  of  all  the  masonry, 
chimney,  boiler,  and  engine  house,  etc.,  be  P^.  Let  the  inter- 
est on  the  price  of  boiler  and  engine  be  x  per  cent.  The  inter- 
est yearly  is  then 

ot(^  +  ^')- 

Let  the  interest  upon  the  cajoital  P^  be  y  per  cent.  Then  we 
have  yearly  for  this 

y    p 

100  ^'' 

Let  the  engine  work  z  hours  per  year,  and  each  hour  con- 
sume B  kilograms  of  fuel  at  a  price  of  P3  per  kilogram.  Then 
the  yearly  expenditure  for  fuel  is 

zBP,. 

If  the  price  for  attendance  is  A  for  each  hour  for  each  horse 
power,  and  for  lubrication  per  year  is  A^,  and  yearly  repairs  A:^, 
we  have  the  yearly  expenditure 

sA  +  NuA^  +  A,. 

The  total  expenditure  per  year  is  then 

zA  +  N^A,  +  A,  +  zBP,  +  i^P,  +  ^(P  +  P,),       ■ 

and  the  expenditure  per  hour  for  each  horse  power  is 

,    _  gA  4-  N,,Ar  +  A,  +  zBP,  +  irP,  +  m(P+  PQ 


604  THERMODYNAMICS. 

The  expenditure  per  hour  for  each  horse  power  is  then,  in 
general,  less, 

1.  The  greater  the  number  of  working  hours  per  year.  This 
is  evident,  as  for  many  interruptions  a  large  amount  of  fuel  is 
wasted. 

2.  The  greater  Nu,  or  the  larger  the  engine.  It  is  also  evi- 
dent that  two  or  more  engines  which  give  the  same  work  as 
one  are  more  costly  in  construction  and  maintenance  than  the 
single  one.  When,  indeed,  the  size  passes  a  certain  limit,  the 
difficulties  of  construction  may  be  so  great  that  two  or  more 
may  be  cheaper  than  one. 

3.  The  higher  the  temperature  of  the  feed  water.  The  heat- 
ing of  the  feed  water,  the  importance  of  which  we  have  shown 
on  page  564,  is  a  general  practice.  For  this  purpose  the 
chimney  gases  are  used,  or  the  heat  of  the  escaping  cylinder 
steam. 

4.  The  greater  the  mean  effective  pressure  pi  and  the  less  the 
mean  back  pressure  p,„.  The  boiler  pressure  should,  there- 
fore, be  high  (6  to  8  atmospheres).  The  higher  this  pressure, 
the  less  is  the  advantage  of  condensation. 

5.  The  greater  the  velocity  of  the  engine.  But  this  has  evi- 
dently a  limit.  Radinger  has  shown  that  for  a  certain  degree 
of  fill  and  a  certain  pressure,  the  usually  received  mean  veloc- 
ity can  be  exceeded  without  danger  of  irregular  action.  The 
motive  force  at  the  beginning  of  each  stroke  must  be  so  great 
as  to  be  able  to  overcome  the  friction  of  the  piston  and  the 
inertia  of  the  moving  masses,  such  as  piston,  piston  and  con- 
necting rods.  This  governs  the  extent  of  compression  (the  ad- 
vantage of  which  has  been  referred  to)  as  also  the  amount  of 
lead. 

Hrabak  (see  Grashof's  Resultate  der  mechanischen  Warme- 
theorie)  gives  for  the  following  useful  deliveries  the  correspond- 
ing mean  velocities  : 


Nu  =  l 

10 

20 

45 

80 

150 

30c  =  30 

35 

40 

45 

50 

55 

Under  the  assumption  that  x  —  lQ,  %—  3600,  D  =^1B  and 
Pz  =  30  cents  about,  the  coefficient  of  fill  ei  has  been  calcu- 
lated for  different  sizes  and  pressures,  when  the  quantity 


CALCULATION  OF  A  STEAM  ENGINE. 


605 


is   a   minimum.     Tliis  quantity  is  the  principal  yearly  cost. 
The  following  table  gives  the  degree  of  fill  e,  : 


I 

engine  wit 

:hout  Cone" 

lensation. 

Condensing  Engines. 

N,,^ 

7 

20 

60 

180 

7        20        60        180 

i^i  =  2 

— 

— 

— 

— 

0.33    0.30     0.25     0.23 

i^:  =  3 

0.41 

0.40 

0.39 

0.38 

0.30    0.25     0.30     0.20 

i^i  =  4 

0.33 

0.32 

0.31 

0.30 

0.25    0.22     0.20     0.15 

B=  6 

0.30 

0.25 

0.23 

0.20 

0.24    0.20     0.18     0.13 

We  see,  therefore, 

a.  That  the  coefficient  of  fill,  for  the  same  delivery  N,,  and 
the  same  steam  pressure  2h  is  less  for  condensing  engines  than 
for  non-condensing.  That,  however,  the  coefficient  for  both 
systems  is  more  nearly  the  same  as  the  pressure  increases. 

h.  Both  condensing  and  non-condensing  engines  have  a  less 
coefficient  of  fill,  the  greater  the  useful  delivery  Nu  and  the 
greater  the  pressure  p^. 

For  a  non-condensing  engine,  for  example,  of  7  horse  power, 
it  v/ould  be,  under  the  assumed  conditions,  not  advantageous, 
for  a  pressure  of  4  atmospheres,  to  have  the  degree  of  fill  of 
the  cylinder  ei  greater  or  less  than  0.33. 

Calculation  of  a  Projected  Steam  Engine. — Let  us  now  conclude 
by  showing  how  to  proceed  in  order  to  find  the  dimensions  of 
the  more  important  parts  of  a  steam  engine  of  any  required 
useful  horse  power.  We  cannot  find  these  dimensions  directly 
from  the  preceding  formulae,  but  we  will  show  how  by  their  aid 
we  can  find  approximate  values,  and  then,  from  these  values 
can  find  the  more  exact  pro23ortions.  Let  us  take  an  example 
which  will  illustrate  the  general  method  of  procedure. 

EXAMPLE. 

Required  to  build  a  steam  engine,  working  expansively,  -wliose  effective  de- 
livery shall  be  60  horse  power. 

We  assume  that  water  is  abundant,  and  henee  the  engine  may  be  a  condensing 
engine.  We  also  assume  that  the  work  required  of  the  engine  is  of  such  char- 
acter that  a  eoeificient  of  irregularity  of  5  =  -gV  will  be  sufficient.  What  dimen- 
sions must  we  give  the  engine  ?  what  amount  of  water  and  fuel  is  required  per 
hour  ?  etc. 


606  THERMODYNAMICS. 

If,  instead  of  the  effective  delivery,  the  indicated  delivery  had  been  given,  our 
formulfB  would  enable  us  to  find  with  ease  the  dimensions  of  the  various  parts, 
and  then  we  could  estimate  the  resistances  owing  to  the  motion  of  these  parts. 
The  work  corresponding  to  these  resistances,  deducted  from  the  indicated  de- 
livery, would  then  give  at  once  the  effective  delivery.  We  see  at  once  that  the 
indicated  delivery  must  be  much  greater  than  the  actual.  We  have  called  the 
quotient  of  the  actual  by  the  indicated  delivery 


mi  — 
the  indicated  efficiency, 


mi  —  -.J-  — 


This  efficiency  m^-  we  must  seek  to  determine  by  experiment 
and  calculation.  It  is,  of  course,  more  for  large  engines  and 
less  for  smaller  ones.  Also  for  the  same  delivery  it  is  some- 
what less  for  expansion  engines  than  for  full  pressure,  and  for 
condensing  engines  least  of  all. 

According  to  Grashof,  we  have 

1.  For  non-condensing  engines  without  expansion,  when 
a.  Nu  is  from  5  to  25  horse  power, 

N,,  +  35 


'  ~  iv;  +  50  * 

h.  N^i,  from  25  to  80  horse  power, 

_  iV;, +  75 
""'^■-JV, +  100* 

2.  For  non-condensing  engines  tvith  expansion, 
a.  Nu  from  10  to  40  horse  power, 


rrii^ 

2^u  + 

32 
50  " 

>.  Nu  from  40  to  100  horse 

power 

5 

m,.-  = 

Nu  + 

72 

iV„, +  100 


CALCULATION'  OF  A   STEAM  ENGINE.  607 

3.  For  condensing  and  expansion  engines. 
a.  JSfu  from  15  to  46  liorse  power, 


N.,  +  50  • 
(6.)  iV^,  from  46  to  180  liorse  power, 

JV, +  86 
N^  +  130  • 


rrii 


From  these  formulae  tlie  following  table  is  calculated,  wMch 
contains  values  rather  too  small  than  too  large  : 

iV;,  =  5     10     15    20    25    30    40 

1  0.727   0.750   0.769   0.786   0.800   0.808   0.821 

2  ....   0.700   0.723   0.743   0.760   0.775   0.800 

3   0.631   0.657   0.680   0.700   0.733 

N,,  =     50    60  80   100    120   150   180 

1  0.833  0.844  0.861       

2  0.813  0.825  0.844  0.860       

3  0.756  0.768  0.790  0.809      0.824      0.843      0.858 

Since  our  engine  is  a  condensing  engine  with  expansion,  we 
have  from  the  table  m,-  =  0.768. 


Hence 


0.768  =  #  =  «»,  or 


^i  =  Qi^gg-  =  78  horse  power. 

Let  the  mean  steam  pressure  of  the  driving  steam  he  pi  —  4 
atmospheres,  and  the  coefficient  of  fill  be  e^  =  0.2,  as  given  from 
the  table,  page  605.  Further,  from  page  604  let  the  velocity  be 
given  by 

30c  =  47,     or    c  =  1.567  meters. 


608  THEBMOD  YNAMIGS. 

Then  from  page  586 

.r      10334      ^ 

From  the  table,  page  585,  we  have  for  pi,  for  ej  =  0.20,  and 
Pi  =  4  atmospheres, 

Pi  =  1.881. 
Hence 

Fr  =    75^-     _        75  X  78       _ 

10334pi  ~  10334  x  1.881  "  ' 

and  since  c  =  1.567  meters,  we  have  for  the  cross-section  F  of 
the  cylinder,  not  including  that  of  the  rod, 

F  =  ^f|i  =  0.192  sq.  meters. 

Since  the  velocity  of  the  piston  (c)  is  1.567  meters,  we  must 
give  to  the  steam  passages,  according  to  page  582,  a  cross-sec- 
tion of  about  yV^i  of  the  cylinder  cross-section.  We  may  make 
them  4  to  5  times  as  long  as  broad. 

If  we  make  the  diameter  of  the  piston  rod  jVth  of  that  of  the 
piston,  we  have  for  d 

4--i^(TV^)^  =  i^-0.192; 

hence 

d  =  0.498  =  diameter  of  the  cylinder, 
and 

0.0498  meters  =  4.98  cm.  =  diameter  of  piston  rod. 

The  stroke  s  of  the  piston  is  best  given  by 

^  =  2.8  -  d, 
d 

hence 

^-2.8-0.498  =  2.302, 

or 

.9  =  1.146  meters. 


CALCULATION  OF  A   STEAM  ENGINE.  609 

We  can  now  find  tlie  weight  Sr  of  the  fly  wheel  rim.     Sup- 
pose we  make  the  connecting  rod  5  times  as  long  as  the  crank, 

T 

then  7-  =  i,  and  from  page  593, 


Sr  =  2.273  '  '  ^    -^^ 


RJ  v^S  ' 

If  we  make  B  four  times  r,  in  which  case  the  mean  diameter 
B  of  the  fly  wheel  rim  will  be  4  x  1,146  =  4.584  meters,  we 
have,  since  Xj  =  75  x  78  =  5850  meter-kilograms,  and  d  =  Jg. 

a      o  o^Q       1       5850  X  32  ,  ., 

Sr  —  2,273  X  Jg  X 2 kilograms. 

For  V  we  have 

Vc      3,1416  X  1.567      „  ,^^       ^ 
v=  -^  = ^ =  2,462  meters. 

Hence 

Sr  =  4390  kilograms. 

Since  our  engine  works  expansively,  and  the  coefficient  of  fill 
is  0.20  =  -^,  the  above  weight  must  be  multiplied  by  a  coefficient 
given  by  the  equation,  page  592, 

0-77  +  0,23  X  5  -  0.017  x  25  =  1.495. 

Hence  the  weight  of  the  fly-wheel  rim  is 

Sr  =  4390  X  1.495  =  6563  kilograms. 

If  we  make  the  arms  y  0  ths  of  the  weight  of  the  rim,  they  will 
weigh  yV  ^  6563  — - 1968.9,  or  in  round  numbers, 

weight  of  fly  wheel  —  1970  kilograms. 

The  dimensions  of  the  rim  and  arms  can  be  easily  found  from 
the  formulae  of  page  594, 

We  can  now  determine  the  diameter  of  the  fly  wheel  journal. 
39 


610  THEBM0DYNAMIC8. 

If  we  denote  it  by  c?i,  we  liave,  according  to  Morin, 


02  i  /  — •*  centimeters, 


u 


where  u  is  tlie  number  of  revolutions  per  minute. 
For  u,  we  liave  from  page  194, 

30c      30  X  1.567       ,-,  ^„ 
^==V^      1.146       ^^^-Q^^ 
hence 


c?i  =  20  \/  -^  =  20/^/  1.902  =  24.8  centimeters. 

If  we  make  the  diameter  of  the  shaft  26.8  centimeters,  and 
make  it  3  meters  long,  since  the  weight  of  1  cubic  meter  of  cast 
iron  is  7400  kilograms,  the  weight  of  the  shaft  is 

3.1416   X  (0.268)2         ^  rr^nn         1 0KA  Tl 

-^ X  3  X  7400  =  1250  kilograms. 

The  total  weight  of  fly  wheel  and  shaft  is  then 

8=8y,  +  1.^8r  =  1250  +  6563  +  1970  =  9783  kil. 

The  mean  effective  pressure  ps  required  for  overcoming  the 
friction  of  the  fly  wheel  shaft  is,  from  page  596, 

.        i?«- 0.00002^—. 
In  the  present  case 

Ps=  0.00002  ^^^^^^,^^^  =  o.m. 

For  the  mean  pressure  (effective)  pt  required  to  overcome  the 
friction  of  the  piston,  piston  rod,  etc.,  and  to  work  the  feed 
pump,  we  have  (page  596) 

0.0227       0.0227      ^n^^ 
^^=-T-- a498=^-^^^' 


CALCULATION  OF  A   STEAM  ENOINE.  611 

Hence  tlie  mean  effective  pressure  p^  of  the  engine  running 
loose,  is,  ivlieii  non-condensing, 

Pm  =  0.170  +  0.046  =  0.216. 

For  a  condensing  engine  we  must  add  a  term  which  includes 
the  working  of  the  cold  water  and  air  pumps  (page  597).  Since, 
however,  the  steam  weight  D  used  by  the  engine  per  hour  oc- 
curs in  this  term,  we  must  first  find  D. 

From  page  600,  we  have  for  D 

D  2=  120 Fsu  [(e  +  e^)  y^  -  ey^]  +  4:50dVp7, 

where  yi  is  the  weight  of  1  cubic  meter  of  steam  at  the  pressure 
of  4  atmospheres,  which  from  Table  XL  is  2.23  kilograms,  y^  = 
0.264,  e  =  0.07  (page  576),  e,  =  0.20,  s  =:  1.146  and  it  =  41.02. 
Hence 

Z>  =  120  X  0.192  X  1.146  x  41.02  [(0.07  +  0.20)  2.23  -  0.07  x 
0.264]  +  450  X  0.498  VLSST  =  632.47  +  307.35  =  939.82  kilgrs. 

Hence  (page  599), 

^ ™  =  (0-0°3''  +  0.045) igo^nmf xfue  X  41.02  ' 
If  we  take  h  =  2  meters,  we  have 

2^'r,^  =  (0.006  +  0.045)  ?|?^  =  0.051  x  0.868  =  0.044. 

Hence 

p^  =  0.216  +  0.044  =  0.260. 

From  page  600  we  have  the  effective  pressure  which  gives 
the  useful  work 

^       g^^^      1.881-6.260  _^g 
-^"  ~     1.13    ~         1.13 

Hence  iV„  (page  600)  is 

^  10334x0.192x1.567x1.435  ^  gg^^^  ^^^^^  ^^^^^^ 
75 


612  THERMODYNAMICS, 

The  result  coincides  then  so  exactly  with  the  required  power 
of  60  horse  power  that  another  and  closer  computation  is  not 
necessary.     This  is  principally  not  only  because  we  have  taken 

y  =  I,  but  also  have  made  -^  =  |.     If,  for  example,  we  had 

taken  -^  =  |,  the  fly  wheel  would  have  been  much  lighter,  and 

hence  the  work  required  for  its  motion  less.  "We  should  then 
have  found  for  N^  a  greater  value  than  60  horse  power.  In 
such  case  the  area  of  the  piston  would  have  to  be  reduced 
somewhat  in  order  to  obtain  the  desired  result. 

Let  us  determine  now  the  dimensions  of  the  condenser  and 
pumps. 

From  page  601,  the  volume  of  the  condenser  is  (7  =  -^  to  -^ 

Fs 
Let  us  take  then   G  =  i^r-^  ,  then 
3.5 


^      0.192  X  L146      ^^„^      ,. 

(J  —  ^^ =  0.063  cubic  meters. 

o.o 

Let  the  cold  water  pump  be  single  acting.     The  volume  V^ 
of  the  same  is,  from  page  602, 


64000ii 
or  taking  n  —  20, 

..         20x939.82        .^^^^      ,.         , 
^^  =  54000  X  41.02  =  ^'^^^^  "^^^'  ^"*"''' 


If  we  make  the  stroke  of  the  pump  one  half  that  of  the  cyl- 
inder, we  have  for  the  cross-section  F^, 


F,^  =  0.0085, 


F,  X  0.573  =  0.0085, 
F,  =  0.0148. 


CALCULATION  OF  A   STEAM  ENGINE.  613 

Hence  we  have  tlie  diameter  d^ 

^  =  0.0148, 

or 

c^2  =  0.138  meters  =  13.8  centimeters. 

If  we  denote  tlie  volume  described  by  tlie  piston  of  tlie  air 
pump  per  stroke  by  V2,  we  can,  in  general,  take 

^'  =  4  to  4.5. 
Taking  tlie  first  value, 

V2  =  0.0085  X  4  =  0.0340  cubic  meters. 

As  soon  as  we  fix  upon  tlie  stroke,  we  can  find  the  diameter. 

The  volume  described  by  the  piston  of  the  feed  pump  per 
stroke,  can  also  be  easily  calculated.  The  feed  water  per  hour 
is  X>  =  939.82  kilograms  =  0.9398  cubic  meters.  If  yVth  of  the 
water  falls  back,  we  have  per  hour  the  water  volume  0.9398  4- 
0.09398  ^  1.0337,  and  per  minute,  0.0172  cubic  meters.  In 
order  to  feed  the  boiler  rapidly,  the  pump  must  furnish  in  this 
time,  3  to  6  times  this  volume.  If  we  say  4  times,  we  have  the 
quantity  per  minute  0.0688  cubic  meters.  If  the  pump  is 
single  acting,  it  makes,  in  41.02  revolutions  of  the  engine,  41.02 
strokes,  or  feeds  the  boiler  41.02  times  per  minute.  Hence  the 
volume  which  the  piston  of  the  feed  pump  must  describe  per 
stroke  is,  since  for  each  time  it  feeds  the  boiler  it  rises  and 
falls 

,?  ^^^^     =  0.00334  cubic  meters. 
41.02  X  ^ 

If  now  (page  604) 

D  =  1B,    or    B  =  \D, 

we  have  for  the  weight  of  fuel  per  hour 

B  =  ??!??  =  134.26  kilograms. 


614  TEEBMOBTNAMICS. 

It  is  thus  assumed  tliat  the  heating  value  of  the  fuel  is  toler- 
ably great,  and  therefore  that  good  hard  coal  is  used. 

"We  can  now  recapitulate  the  dimensions  of  our  engine  as 
calculated,  or  given. 

1.  Diameter  of  cylinder  {d) 0.498  meters. 

2.  Length  of  stroke  {s) 1.146 

3.  Cross-section  of  steam  passages 0.0101  sq.  m. 

4.  Mean  velocity  of  piston  (c) 1.567  meters. 

5.  Coefficient  of  fill  {e,) 0.20 

6.  Revolutions  per  minute  {u) 41.02 

7.  Diameter  of  the  piston  rod 4.98  centim. 

8.  Eatio  of  length  of  crank  to   connecting 

,    r  1 

^^^R--- 5 

9.  Length  of  crank  (r)  =  ^ 0.573  meters. 

A 

10.  Weight  of  fly  wheel  rim 6563  kilogrs. 

11.  Weight  of  fly  wheel  arms 1970 

12.  Weight  of  fly  wheel  shaft 1250 

13.  Diameter  of  journals 24.8  centim. 

14.  Diameter  of  shaft 26.8         " 

15.  Steam  consumption  per  hour   939.82  kilogrs. 

16.  Condensing  water  per  hour  20  x  939.82  . . .  .1879.64      " 

17.  Consumption  of  coal  per  hour.  .134.26  kil.=  18.7964  cub.  m. 

18.  Volume  of  condenser  (0) 0.063 

19.  Volume    described   by   piston   of    cold 

water  pump  per  stroke 0.0085       " 

20.  Volume  described  by  the  piston  of  the 

air  pump  per  stroke 0.0340       " 

21.  Volume  described  by  the  piston  of  the 

feed  pump  per  stroke 0.00334     " 


EXAMPLES  FOR  PRACTICE. 


1.  "What  is  the  pressure  of  saturated  steam  whose  temperature  is  20'  Fah.  ? 
What  is  the  temperature  for  a  pressure  of  10  atmospheres  'i  What  is  the  pressure 
for  140   C?  ^ 

2.  What  is  the  mean  specific  heat  of  water  between  10'  and  25'  C.  ?  Between 
25°  and  70^  Fah.  ?  How  much  heat  is  required  to  raise  10  lbs.  of  water  from  60' 
to  80'  Fah.  ?  How  much  to  raise  2  kilograms  from  30'  to  70'  C.  ?  What  is  the 
specific  heat  of  water  at  212'  Fah.  ?    At  140'  C.  ? 

3.  How  much  heat  is  required  to  convert  2  lbs.  of  water  at  30'  Fah.  into 
saturated  steam  at  300'  Fah.  ?  How  much  to  convert  1  kilogram  of  water  at  10° 
C.  into  saturated  steam  of  120'  C.  ? 

4.  How  much  heat  is  required  to  vaporize  1  lb.  of  water  at  300°  Fah.  into 
steam  of  the  same  temperature  ?  What  is  the  pressure  ?  What  is  the  outer 
work? 

5.  What  is  the  volume  of  a  quantity  of  steam  and  water  at  212°  Fah.  whose 
weight  is  1  lb.,  and  which  consists  of  0.2  lb.  of  steam  and  0.8  lb.  of  water  ? 

6.  One  pound  of  steam  and  water  has  a  temperature  of  250°  Fah.,  of  which  2 
cubic  feet  are  steam.  How  much  does  the  steam  weigh  ?  How  much  does  the 
water  weigh  ? 

7.  What  is  the  specific  volume  of  steam  at  80°  C.  ?    At  240'  Fah.  ? 

8.  What  is  the  outer  work  performed  in  converting  2  lbs.  of  water,  at  250° 
Fah.  into  steam,  under  a  constant  pressure  equal  to  the  steam  tension  ?  What 
is  the  steam  tension  ?    What  is  the  steam  volume  ? 

9.  What  is  the  density  of  steam  at  4  atmospheres'  pressure  ?  What  is  its  tem- 
perature ?  If  the  volume  is  3  cubic  feet,  what  volume  and  weight  of  water  were 
necessary  to  form  it  ? 

10.  How  many  heat  units  must  be  imparted  to  1  lb.  of  saturated  steam,  in 
order  to  keep  it  all  saturated  while  it  expands,  performing  work,  till  the  tempera- 
tui-e  sinks  1°  Fah.  ?  If  the  initial  temperature  is  222°  Fah.,  what  is  the  initial 
volume  ?  What  is  the  final  volume  ?  The  initial  pressure  ?  The  final  pressure  ? 
The  outer  work  done  ? 

11.  How  many  heat  units  must  be  imparted,  as  before,  to  1  lb.  of  saturated 
steam  when  it  expands,  performing  work,  from  5  atmospheres  down  to  1  atmos- 
phere ?  What  are  the  initial  and  final  volumes  ?  Initial  and  final  temperatures  ? 
Work  done  during  expansion  ? 

12.  A  full-pressure  non-condensing  engine  has  a  stroke  of  3  feet,  cross-sec- 
tion of  piston,  1.5  sq.  feet.  The  steam  pressure  is  5  atmospheres,  and  it  makes 
25  revolutions  per  minute.  What  is  the  theoretical  work  per  second,  and  how 
much  heat  is  required  ? 

615 


616  THERMODYNAMICS. 

13.  If  a  condensing  engine  forces  3  cubic  feet  of  steam  at  a  pressure  of  n,th 
of  an  atmosphere  into  the  condenser,  what  work  is  necessary,  and  liow  much  heat 
is  taken  from  tlie  steam  V 

14.  What  work  is  performed  by  the  adiabatic  expansion  of  1  lb.  of  saturated 
steam  from  4  atmosplieres  to  1  atmosphere  ?    How  mucli  steam  is  condensed  ? 

15.  What  would  the  work  be  if,  to  start  with,  we  had  only  water  and  no  steam? 
How  much  steam  would  be  formed  ? 

16.  If  1  lb.  of  a  mixture  of  0.8  lb.  steam  and  0.2  lb.  water,  expands  adiabati- 
cally  from  8  atmospheres  down  to  1  atmosphere,  what  is  the  work  performed  ? 
What  is  the  initial  volume  'i  Final  volume  ?  Heat  disappearing  ?  How  much 
steam  is  condensed  ? 

17.  If  a  mixture  of  10  lbs.  is  composed  of  0.8  steam  and  0.2  water,  and  has  a 
pressure  of  1.5  atmospheres,  what  will  be  the  amount  of  steam  and  water  when 
the  mixture  is  cooled,  under  constant  volume,  until  the  pressure  is  f'„th  of  an  at- 
mosphere ?    What  amount  of  heat  must  be  abstracted  ? 

18.  A  boiler  has  170  sq.  feet  of  hcatmg  surface,  and  contains  300  cubic  feet, 
of  which  0.6  are  water  and  the  rest  steam.  In  ordinary  use,  the  boiler  generates 
per  hour  5  lbs.  of  steam  for  every  sq.  foot  of  heating  surface,  of  5  atmospheres' 
tension.  In  how  many  minutes  will  the  pressure  rise  to  10  atmospheres,  the  tem- 
perature of  the  feed  water  being  60'  Fah.  ? 

19.  If  a  vessel  containing  G  lbs.  of  pure  saturated  steam,  at  1.5  atmospheres, 
communicates  with  another  containing  25  6^  lbs.  of  a  mixture  of  water  and 
steam,  at  i\.th  atmosphere,  of  whioh  0.03  of  a  pound  are  steam,  what  is  the  con- 
dition of  the  mixture  after  the  cock  is  opened  ? 

20.  Given  10,000  lbs.  of  feed  water  at  190"  Fah.  evaporated  at  70  lbs.  steam 
gauge  pressure,  and  the  steam  containing  2.75  per  cent,  of  moisture.  Find  iho 
heat  units  required  for  evaporation.  Also  suppose  1200  lbs.  of  coal  were  con- 
sumed, find  the  efficiency  of  the  boiler.     Efficiency  =  =j '—-. — q — ~ -. —  ; 

Tlieoretical   Evaporation 
also,  iind  the  equivalent  evaporation  at  and  from  212°  Fah. 

21.  Suppose  a  calorimeter  used  for  determining  the  moisture  of  steam,  which 
not  only  condenses  but  retains  the  steam  and  spray  admitted. 
Further,  let 

W  =  TFi  +  C '  TTj  =  the  sum  of  the  original  weight  of  condens- 
ing water,  and  the  product  of  the  specific  heat  of  the  testing  vessel  by  its  weight. 

IV  =  weight  of  mixture  of  steam  and  spray. 

X  =  weight  of  steam  in  the  mixture. 

IV  —  X  =  weight  of  spray  in  the  mixture. 

r  =  total  latent  heat  of  steam. 

p.,  =  steam  gauge  pressure  (  =  excess  above  atmosphere). 

I)  =  Qs'  ~  q  =  (ts"  —  t^  nearly)  =  difference  between  the  heat  of  liquid  at 
temperature  of  the  steam  and  at  ifinal  temperature  of  water  in  condensing  vessel. 

d  —  q  —  q^  ={t"  —  t,°  nearly)  =  difference  between  heat  of  liquid  at  final 
temperature  of  the  water,  and  at  initial  temperature  of  condensing  water.  Prove 
that  if  no  external  work  is  done  while  the  steam  is  condensing,  the  percentage  of 
moisture  is 

w-x        .  Wiq-q,)-iciq,-  q) 


By  experiment,  we  find  W^  =  5.796  lbs.,  TF,  =  3.858  lbs.,  G'  =  0.11,  zv  = 
0.25  lbs.,  ps  =  365  ibs.,  #,  =  65.5°  Fah.,  t  =  108.3°  Pah.,  what  is  the  per- 
centage of  moisture  in  the  steam  ? 

22.  In  a  surface  condenser  the  water  enters  with  a  temperature  of  60"  Fah., 
and  departs  at  80"  Fah.  The  mean  temperature  of  the  condenser  is  115°  Fah. 
How  much  more  condensing  v/ater  than  steam,  by  weight,  must  be  used  ? 


EXAMPLES  FOB  PRACTICE.  617 

23.  An  engine  using  steam  of  5  atmospheres  has  a  jet  condenser  in  which  the 
average  pressure  is  0.1  atmosphere.  Tlie  cooling  water  has  a  temperature  of 
60°  I'ah.     How  much  more  water  than  steam  must  be  used  ? 

24.  A  boiler  contains  steam  at  a  pressure  of  5  atmospheres.  When  the  safety- 
valve  is  opened,  what  is  the  velocity  of  efElux,  disregarding  friction,  and  sup- 
posing the  steam  to  be  diy  ?    How  much  steam  is  condensed  during  efflux  ? 

25.  What  diameter  should  the  safety  valve  of  a  steam  boiler  have,  which 
generates  per  hour  500  lbs.  of  steam  at  5  atmospheres,  for  20-fold  security  ? 

26.  Hot  water  is  allowed  to  flow  from  the  test  cock  of  a  boiler  under  the  press- 
ure of  5  atmospheres.  What  is  the  specific  steam  weight  at  the  orifice  ?  With 
what  velocity  does  the  mixture  issue  ':"  What  is  the  discharge  per  second  ?  How 
much  steam  and  water  are  contained  in  the  mixture  ? 

37.  The  steam  pressure  in  a  boiler  is  5  atmospheres,  the  height  of  suction  8 
feet.  The  condensing  chamber  is  at  the  water  level.  The  .engine  uses  20  lbs. 
of  steam  per  minute.  What  should  be  the  area  of  the  mouthpiece  of  a  Gitfard 
injector  ?  What  of  the  suction  pipe  and  the  feed  pipe  when  the  feed  water  has 
a  temperature  of  60'  Fah.,  and  the  mixture  of  water  and  steam  120°  Fah.  ? 

28.  If  one  pound  of  dry  saturated  steam  at  3  atmospheres  expands  in  vacuo 
down  to  1  atmosphere,  what  is  the  temperature  ?  How  many  degrees  must  satu- 
rated steam  of  one  atmosphere  be  heated  under  constant  pressure,  in  order  that 
for  the  same  temperature  it  may  have  the  same  volume  ? 

29.  An  engine  works  with  superheated  steam  of  5  atmospheres  and  tempera- 
ture 360"  Fah.  What  is  the  expansion  ratio  when  the  steam  at  the  end  of  expan- 
sion is  just  in  the  saturated  condition  ? 

30.  A  vessel  contains  one  poimd  of  piire  saturated  steam  at  5  atmospheres. 
Let  the  steam  in  this  vessel  expand  into  another  in  which  is  a  vacuum,  whose 
volume  is  4  times  as  large.  What  is  the  final  pressure  and  temperature  ?  And 
is  the  steam  superheated  ? 

31.  If  saturated  steam  of  5  atmospheres  expands  under  constant  temperature 
down  to  1  atmosphere,  Avhat  is  the  heat  imparted  ?     The  outer  work  done  ? 

32.  Suppose  we  have  10  lbs.  of  saturated  steam  of  5  atmospheres.  What  is 
the  heat  required  to  generate  it  ?  How  much  heat  is  required  to  generate  the 
same  volume  of  superheated  steam  of  the  same  pressure  ? 

33.  In  a  hot-air  engine  the  heat  furnished  per  hour  to  the  air  is  6200  heat 
xinits,  while  in  the  same" time  10  lbs.  of  coal  are  consumed,  whose  heating  value  is 
700  heat  units.     What  is  the  efficiency  of  the  furnace  ? 

34.  The  boiler  of  an  expansion  engine,  which  furnishes  steam  of  5  atmospheres, 
vaporizes  per  hour,  for  every  horse  power,  60  lbs.  of  water,  and  requires  for  this 
10  lbs.  of  coal,  whose  heating  power  is  700  heat  units.  What  is  the  boiler 
efficiency  ? 

35.  What  would  be  the  delivery  of  a  perfect  steam  engine  using  per  hour  200 
lbs.  of  steam  of  10  atmospheres  ? 


THEBMOD  TNAMIG8. 


619 


TABLE  I. 


FORCE    OF  STEAM  FOR  TEMPERATURES   FROM  — . 
ACCORBING  TO   REGNAULT. 


i 

3 

a 

TENSION 

JF  STEAM. 

1 

1 

TENSION 

OF  STEAM. 

In  Centimeters. 

In  Atmospheres. 

In  Centimeters. 

In  Atmospheres. 

-33^ 

0.0320 

0.0004 

+14° 

1.1908 

0.016 

31 

0.0352 

0.0005 

15 

1.2699 

0.017 

30 

0.0386 

0.0005 

16 

1.3536 

0.018 

29 

0.0424 

0.0006 

17 

1.4421 

0.019 

28 

0.0464 

0.0006 

18 

1.5357 

0.030 

37 

0.0508 

0.0007 

19 

1.6346 

0.033 

26 

0.0555 

0.0007 

20 

1.7391 

0.033 

25 

0.0605 

0.0008 

21 

1.8495 

0.034  ' 

•24 

0.0660 

0.0009 

22 

1.9659 

0.036 

23 

0.0719 

0.0009 

23 

2.0888 

0.038 

22 

0.0783 

0.0010 

24 

2.2184 

0.039 

21 

0.0853 

0.0011 

25 

2.3550 

0.031 

20 

0.0927 

0.0012 

26 

2.4988 

0.033 

19 

0.1008 

0.0013 

27 

2.5505 

0.034 

18 

0.1095 

0.0014 

28 

2.8101 

0.037 

17 

0.1189 

0.0015 

29 

2.9782 

0.039 

16 

0.1290 

0.0017 

30 

3.1548 

0.042 

15 

0.1400 

0.0018 

31 

3.3406 

0.044 

14  ■ 

0.1518 

0.0020 

32 

3.5359 

0.047 

13 

0.1646 

0.0022 

33 

3.7411 

0.049 

12 

0.1783 

0.0024 

34 

3.9565 

0.052 

11 

0.1933 

0.0025 

35 

4.1827 

0.055 

10 

0.2093 

0.0027 

36 

4.4201 

0.058 

9 

0.2267 

0.0030 

37 

4.6691 

0.061 

8 

0.2455 

0.0032 

38 

4.9302 

0.065 

7 

0.2658 

0.0035 

39 

5.2039 

0.068 

6 

0.2876 

0.0038 

40 

5.4906 

0.072 

5 

0.3113 

0.0041 

41 

5.7910 

0.076 

4 

0.3368 

0.0044 

42 

6.1055 

0.080 

3 

0.3644 

0.0048 

43 

6.4346 

0.085 

2 

0.3941 

0.0052 

44 

6.7790 

0.089 

1 

0.4263 

0.0056 

45 

7.1391 

0.094 

0 

0.4600 

0.0061 

46 

7.5158 

0.099 

+  1 

0.4940 

0.0065 

47 

7.9093 

0.104 

2 

0.5302 

0.0070 

48 

8.3204 

0.109 

3 

0.5687 

0.0075 

49 

8.7499 

0.115 

4 

0.6097 

0.0080 

50 

9.1983 

0.121 

5 

0.6534 

0.0086 

51 

9.6661 

0.137 

6 

0.6998 

0.0092 

52 

10.1543 

0.134 

7 

0.7492 

0.0099 

53 

10.6636 

0.140 

8 

0.8017 

0.0107 

54 

11.1945 

0.147  • 

9 

0.8574 

0.011 

55 

11.7478 

0.155 

10 

0.9165 

0.013 

56 

12.3244 

0.163 

11 

0.9792 

0.013 

57 

12.9351 

0.170 

12 

1.0457 

0.014 

58 

13.5505 

0.178 

13 

1.1162 

0.015 

59 

14.3015 

0.187 

620 


THERMOD  TNAMICS. 


6 

■   1 

? 

f 

TENSION  OF  STEAM. 

3 
1 

TENSION  OF  STEA3I. 

^ 
& 

In  Centimeters. 

In  Atmospheres. 

In  Centimeters. 

In  Atmospheres. 

+  60' 

14.8791 

0.196 

113 

118.861 

1.564 

61 

15.5839 

0.205     1 

114 

122.847 

1.616 

62 

16.3170 

0.215 

115 

126.941 

1.670 

63 

17.0791 

0.225 

116 

131.147 

1.726 

64 

17.8714 

0.235 

117 

135.466 

1.782 

65 

18.6945 

0.246 

118 

139.902 

1.841 

66 

19.5496 

0.257 

119 

144.455 

1.901 

67 

20.4376 

0.267 

120 

149.128 

1.962  . 

68 

21.3596 

0.281 

121 

153.925 

2.025 

69 

22.3165 

0.294 

122 

158.847 

2.091 

70 

23.3093 

0.306 

123 

163.896 

2.157 

71 

24.3393 

0.320 

124 

169.076 

2.225 

72 

25.4073 

0.334 

125 

174.388 

2.295 

73 

26.5147 

0.349 

126 

179.835 

2.366 

74 

27.6624 

0.364 

127 

185.420 

2.480 

75 

28.8517 

0.380 

128 

191.147 

2.515 

76 

30.0838 

0.396 

129 

197.015 

2.592 

77 

31.3600 

0.414 

130 

203.028 

2.671 

78 

32.6811 

0.430 

131 

209.194 

2.753 

79 

34.0488 

0.448 

132 

215.503 

2.836 

80 

35.4643 

0.466 

138 

221.969 

2.921 

81 

36.9287 

0.486 

134 

228.592 

3.008 

82 

38.4435 

0.506 

135 

235.373 

3.097 

83 

40.0101 

0.526 

136 

242.316 

3.188 

84 

41.6298 

0.548 

137 

249.423 

3.28? 

85 

43.3041 

0.570 

138 

256.700 

3.378 

86 

45.0344 

0.593 

139 

264.144 

3.476 

87 

46.8221 

0.616 

140 

271.763 

3.576 

88 

48.6687 

0.640 

141 

279.557 

3.678 

89 

50.5759 

0.665 

142 

287.530 

3.783 

90 

52.5450 

0.691 

143 

295.686 

3.890 

91 

54.5778 

0.719 

144 

304.026 

4.000 

92 

56.6757 

0.746 

145 

312.555 

4.113 

93 

58.8406 

0.774 

146 

321.274 

4.227 

94 

61.0740 

0.804 

147 

330.187 

4.844 

95 

63.3778 

0.834 

148 

339.298 

4.464 

96 

65.7535 

0.865 

149 

348.609 

4.587 

97 

68.2029 

0.897 

150 

358.123 

4.712 

98 

70.7280 

0.931 

151 

367.843 

4.840 

99 

73.3305 

0.965 

152 

377.774 

4.971 

100 

76.000 

1.000 

153 

387.918 

5.104 

101 

78.7590 

1.036 

154 

398.277 

5.240 

102 

81.6010 

1.074 

155 

408.856 

5.380 

103 

84.5280 

1.112 

156 

419.659 

5.522 

104 

87.5410 

1.152 

157 

430.688 

5.667 

105 

90.6410 

1.193 

158 

441.945 

5.815 

106 

93.8310 

1.235 

159 

453.436 

5.966 

107 

97.1140 

1.278 

160 

465.162 

6.120 

108 

100.4910 

1.322 

161 

477.128 

6.278 

109 

103.965 

1.368 

162 

489.836 

6.439 

110 

107.537 

1.415 

163 

501.791 

6.603 

111 

111.209 

1.463 

164 

514.497 

6.770 

112 

114.983 

1.513 

|1G5 

527.454 

6.940 

THEBMOD  TNAMIGS. 


621 


1 

TENSION 

DF  STEAM. 

t 

1 

TENSION 

3F  STEAM. 

In  Centimeters. 

In  Atmospheres. 

In  Centimeters. 

In  Atmospheres. 

+  166' 

540.669 

7.114 

199 

1144.746 

15.062 

167 

554.143 

7.291 

200 

1168.896 

15.380 

168 

567.882 

7.472 

201 

1193.437 

15.703 

169 

581.890 

7.656 

202 

1218.369 

16.031 

170 

596.166 

7.844 

203 

1243.700 

16.364 

171 

610.719 

8.036 

204 

1269.430 

16.703 

172 

625.548 

8.231 

205 

1295.566 

17.047 

173 

640.660 

8.430 

206 

1322.112 

17.396 

174 

656.055 

8.633 

207 

1349.075 

17.751 

175 

671.743 

8.839 

208 

1376.453 

18.111 

176 

687.722 

9.049 

209 

1404.252 

18.477 

177 

703.997 

9.263 

210 

1432.480 

18.848 

178 

720.572 

9.481 

211 

1461.132 

19.226 

179 

737.452 

9.703 

212 

1490.222 

19.608 

180 

754.639 

9.929 

213 

1519.748 

19.997 

181 

772.137 

10.150 

214 

1549.717 

20.391 

182 

789.952 

10.394 

215 

1580.133 

20.791 

183 

808.084 

10.633   . 

216 

1610.994 

21.197 

184 

826.540 

10.876 

217 

1642.315 

21.690 

185 

845.323 

11.123 

218 

1674.090 

22.027 

186 

864.435 

11.374 

219 

1708.329 

22.452 

187 

883.882 

11.630 

220 

1739.036 

22.882 

188 

903.668 

11.885 

221 

1772.213 

23.319 

189 

923.795 

12.155 

222 

1805.864 

23.761 

190 

944.270 

12.425 

223 

1839.994 

24.210 

191 

965.093 

12.699 

224 

1874.607 

24.666 

192 

986.271 

12.977 

225 

1909.704 

25.128 

193 

1007.804 

13.261 

226 

1945.292 

25.596 

194 

1029.701 

13.549 

227 

1981.376 

26.071 

195 

1051.963 

13.842 

228 

2017.961 

26.552 

196 

1074.595 

14.139 

229 

2055.048 

27.040 

197 

1097.500 

14.441 

230 

2092.640 

27.535 

198 

1120.982 

14.749 

622 


TEEBMOD  TNAMIC8. 


TABLE  II. 

PRINCIPAL  TABLE  FOR    SATURATED   STEAM. 


1 

2 

3 

4 

5 

6 
Inner  latent 

Steam  pressure 

Temperature 

heat. 

C. 

Heat  of  liquid 

P 

Atmospheres. 

In  millimeters 
of  btirometer. 

Kilograms  per 
sq.  meter. 

t 
(page  37G) 

(pagf  376) 

[?il?S^ 

P 

(Regnault.) 

(Regnault.) 

0.T91AJ 

0.1 

76 

1033.4 

46.21 

46.282 

538.848 

0.2 

152 

2066.8 

60.45 

60.589 

537.584 

0.3 

228 

3100.2 

69.49 

69.687 

520.433 

0.4 

304 

4133.6 

76.25 

76.499 

515.086 

0.5 

380 

5167.0 

81.71 

82.017 

510.767 

0.6 

456 

6200.4 

86.32 

86.662 

507.131 

0.7 

532 

7233.8 

90.32 

90.704 

503.957 

0.8 

608 

8267.2 

93.88 

94.304 

501.141 

0.9 

684 

9300.6 

97.08 

97.543 

498.610 

1.0 

760 

10334.0 

.100.00 

100.500 

496.300 

1.1 

836 

11367.4 

102.68 

103.216 

494.180 

1.2 

912 

12400.8 

105.17 

105.740 

493.310 

1.3 

988 

13434.2 

107.50 

108.104 

490.367 

1.4 

1064 

14467.6 

109.68 

110.316 

488.643 

1.5 

1140 

15501.0 

111.74 

112.408 

487.014 

1.6 

1216 

16534.4 

113.69 

114.389 

485.471 

1.7 

1292 

17567.8 

115.54 

116.269 

484.008 

1.8 

1368 

18601.2 

117.30 

118.059 

483.616 

1.9 

1444 

19634.6 

118.99 

119.779 

481.379 

2.0 

1520 

20668.0 

120.60 

121.417 

480.005 

2.1 

1596 

21701.4 

122.15 

122.995 

478.779 

2.3 

1672 

22734.8 

123.64 

124.513 

477.601 

2.3 

1748 

23768.2 

125.07 

135.970 

476.470 

2.4 

1824 

24801.6 

126.46 

127.386 

475.370 

2.5 

1900 

25835.0 

127.80 

138.753 

474.310 

2.6 

1976 

26868.4 

129.10 

130.079 

473.383 

2.7 

2052 

27901.8 

130.35 

131.354 

473.393 

2.8 

2128 

28935.2 

131.57 

133.599 

471.338 

2.9 

2204 

29968.6 

132.76 

133.814 

470.387 

3.0 

2280 

31002.0 

133.91 

134.989 

469.477 

3.1 

2356 

32035.4 

135.03 

136.133 

468.591 

3.2 

2432 

33068.8 

136.12 

137.347 

467.739 

3.3 

2508 

34102.2 

137.19 

138.341 

466.883 

3.4 

2584 

35135.6 

138.23 

139.404 

466.060 

TEEBMOB  YNAMIC8. 


623 


TABLE  II.— continued. 

PRINCIPAL    TABLE   FOR   SATURATED   STEAJI. 


7 

8 

1               9 

10 

11 

12 

Outer  latent 

Values  of 

heat 

Apv, 

page  393. 

Cub.  metrei=  per 
page  391. 

u 
page  393. 

Difference. 

Kil.  per  cub. 
meter  . 
page  396. 

Difference. 

35.464 

14.5508 

37.03 

0.0687 

36.764 

7.5421 

69.95 

32.92 

0.1326 

0.0639 

37.574 

5.1388 

101.27 

31.32 

0.1945 

619 

38.171 

3.9154 

131.55 

30.28 

0.2553 

608 

38.637 

3.1705 

161.10 

29.55 

0.3153 

600 

39.045 

2.6700 

189.93 

28.83 

0.3744 

591 

39.387 

2.3086 

218.29 

28.36 

0.4330 

586 

39.688 

2.0355 

246.20 

27.91 

0.4910 

580 

39.957 

1.8216 

273.72 

27.52 

0.5487 

577 

40.200 

1.6494 

300.90 

27.18 

0.6059 

572 

40.421 

1.5077 

327.77 

26.87 

0.6628 

569 

40.626 

1.3891 

354.35 

26.58 

0.7194 

566 

40.816 

1.2882 

380.66 

26.31 

0.7757 

563 

40.993 

1.2014 

406.73 

26.07 

0.8317 

560 

41.159 

1.1258 

432.58 

25.85 

0.8874 

557 

41.315 

1.0595 

458.22 

25.64 

0.9430 

556 

41.463 

1.0007 

483.66 

25.44 

0.9983 

553 

41.602 

0.9483 

508.93 

25.27 

1.0534 

551 

41.734^ 

0.9012 

534.03 

25.10 

1.1084 

550 

41.861 

0.8588 

558.94 

24.91 

1.1631 

547 

41.981 

0.8202 

583.72 

24.78 

1.2177 

546 

42.096 

0.7851 

608.34 

24.62 

1.2721 

544 

42.207 

0.7529 

6.32.82 

24.48 

1.3264 

543 

42.314 

0.7234 

657.14 

24.32 

1.3805 

541 

42.416 

0.6961 

681.36 

24.22 

1.4345 

540 

42.515 

0.6709 

705.43 

24.07 

1.4883 

538 

42.610 

0.6475 

729.42 

23.99 

1.5420 

537 

42.702 

0.6257 

753.24 

23.82 

1.5956 

536 

42.791 

0.6054 

776.97 

23.73 

1.6490 

534 

42.876 

0.5864 

800.61 

23.64 

1.7024 

534 

42.960 

0.5686 

824.13 

23.52 

1.7556 

532 

43.040 

0.5518 

847.57 

23.44 

1.8088 

532 

43.119 

0.5361 

870.88 

23.31 

1.8618 

530 

43.196 

0.5213 

894.09 

23.21 

1.9147 

529 

624 


THERMOD  TNAMIC8. 


TABLE  11.— continued. 

PRINCIPAL  TABLE   FOR    SATURATED   STEAM. 


1 

2 

3 

4 

5 

6 
Inner  latent 

Steam  pressure 

Temperature 

heat 

C. 

Heat  of  liquid 

P 

Atmospheres. 

Inmiliimeters 
of  barometer. 

Kilograms  per 
sq.  meter. 

t 
(page  376) 

Q 

(page  3T6) 

(pa?e  393) 
[p  =  .575.40 - 

P 

(Eegnault.) 

(Regnault.) 

0.791M 

3.5 

2860 

36169.0 

139.24 

140.438 

465.261 

3.6 

2736 

37202.4 

140.23 

141.450 

464.478 

3.7 

2812 

38235.8 

141.21 

142.453 

463.703 

3.8 

2888 

39239.2 

142.15 

143.416 

462.959 

8.9 

2964 

40302.6 

143.08 

144.368 

462.224 

4.0 

3040 

41336.0 

144.00 

145.310 

461.496 

4.1 

3116 

42369.4 

144.89 

146.222 

460.792 

4.2 

3192 

43402.8 

145.76 

147.114 

460.104 

4.3 

3268 

44436.2 

146.61 

147.935 

459.431 

4.4 

3344 

45469.6 

147.46 

148.837 

458.759 

4.5 

3420 

46503.0 

148.29 

149.708 

458.103 

4.6 

3496 

47538.4 

149.10 

150.539 

457.462 

4.7 

3572 

48569.8 

149.90 

151.360 

456.829 

4.8 

3648 

49603.2 

150.69 

152.171 

456.204 

4.9 

3724 

50833.6 

151.46 

152.981 

455.595 

5.0 

3800 

51670.0 

152.22 

153.741 

454.994 

5.1 

3876 

52703.4 

152.97 

154.512 

454.401 

5.2 

3952 

53736.8 

153.70 

155.262 

453.823 

5.3 

4028 

54770.2 

154.43 

156.012 

453.246 

5.4 

4104 

55803.6 

155.14 

153.741 

452.684 

5.5 

4180 

56837.0 

155.85 

157.471 

452.123 

5.6 

4256 

57870.4 

156.54 

158.181 

451.577 

5.7 

4332 

58903.8 

157.22 

158.880 

451.039 

5.8 

4408 

59937.2 

157.90 

159.579 

450.501 

5.9 

4484 

60970.6 

158.56 

160.259 

449.979 

6.0 

4560 

62004.0 

159.22 

160.938 

449.457 

6.1 

4636 

63037.4 

159.87 

161.607 

448.943 

6.2 

4712 

64070.8 

160.50 

162.255 

448.444 

6.3 

4788 

65104.2 

161.14 

162.915 

447.938 

6-4 

4864 

66137.6 

161.76 

163.553 

447.448 

6.5 

4940 

67171.0 

162.37 

164.181 

446.965 

6.6 

5016 

68204.4 

162.98 

164.810 

446.483 

6.7 

5092 

69237.8 

163.58 

165.428 

446.008 

6.8 

5168 

70271.2 

164.18 

166.047 

445.534 

6.9 

5244 

71304.6 

164.76 

166.645 

445.075 

lEEBMOD  YNAMIC8. 


625 


TABLE  11.— continued. 

PRINCIPAL    TABLE   FOR   SATURATED   STEAM. 


7 

8 

1               9 

10 

11 

12 

Outer  latent 

Values  of 

heat 

Ap  11, 

page  393. 

u 

Cub.  meters  pei 

kil.. 

page  391. 

page  393. 

Difference. 

Kil.  per  cub. 
meter, 
page  39(5. 

Difference. 

43.269 

0.5072 

917.2 

1.9676 

43.342 

0.4940 

940.3 

23.1 

2.0203 

0.0527 

43.413 

0.4814 

963.2 

22.9 

2.0729 

526 

43.483 

0.4695 

986.1 

22.9 

2.1255 

526 

43.548 

0.4581 

1008.9 

22.8 

2.1780 

525 

43.614 

0.4474 

1031.6 

22.7 

2.2303 

523 

43.677 

0.4371 

1054.2 

22!  6 

2.2826 

523 

43.739 

0.4273 

1076.8 

•      22.6 

2.3349 

523 

43.799 

0.4179 

1099.3 

22.5 

2.3871 

522 

43.859 

0.4090 

1121.7 

22.4 

2.4391 

520 

48.918 

0.4004 

1144.0 

22.3 

2.4911 

520 

43.975 

0.3922 

1166.3 

22.3 

2.5430 

519 

44.030 

0.3844 

1188.5 

22.2 

2.5949 

519 

44.085 

0.3768 

1210.6 

22.1 

2.6467 

518 

44.139 

0.3696 

1232.7 

22.1 

2.6984 

517 

44.192 

0.3626 

1254.7 

22.0 

2.7500 

516 

44.243 

0.3559 

1276.6 

21.9 

2.8016 

516 

44.293 

0.3495 

1298.5 

21.9 

2.8531 

515 

44.343 

0.3433 

1320.3 

21.8 

2.9046 

515 

44.392 

0.3373 

1342.1 

21.8 

2.9560 

514 

44.441 

0.3315 

1363.8 

21.7 

3.0073 

513 

44.487 

0.3259 

1385.4 

21.6 

3.0586 

513 

44.533 

0.3205 

1407.0 

21.6 

3.1098 

512 

44.579- 

0.31.53 

1428.5 

21.5 

3.1610 

512 

44.623 

0.3103 

1450.0 

21.5 

3.2122 

512 

44.667 

0.3054 

1471.5 

21.5 

8.2632 

510 

44.710 

0.3007 

1492.9 

21.4 

3.3142 

510 

44.753 

.  0.2962 

1514.2 

21.3 

3.3652 

510 

44.794 

0.2917 

1535.5 

21.3 

8:4161 

509 

44.836 

0.2874 

1556.7 

21.2 

3.4670 

509 

44.876 

0.2833 

1577.9 

21.2 

8.5178 

508 

44.916 

0.2792 

1599.0 

21.1 

3.5685 

507 

44.956 

0.2753 

1620.1 

21.1 

3.6192 

507 

44.994 

0.2715 

1641.2 

21.1 

3.6699 

507 

45.032 

0.2678 

1662.2 

21.0 

3.7206 

507 

40 


626 


THEBMOD  YNAMIC8, 


TABLE  II.— continued. 

PRINCIPAL   TABLE    FOB    SATURATED    STEAM. 


1 

Atmospheres. 

2 

steam  pressure 

In  millimeters 
of  barometer. 

3 

Kilograms  per 

sq.  meter. 

P 

4 

Temperature 

C. 

t 

(page  376) 

(Regnault.) 

5 

Heat  of  liquid 

q 

(page  376) 

(Eegnault.) 

6 

Inner  latent 

heat 

(page  393) 

[p  =  37.5.40 - 

0.79U.] 

7.00 

7.35 
7.50 

7.75 

5320 
5510 
5700 
5890 

72338.0 

74921.5 
77505.0 

80088.5 

165.34 
166.77 
168.15 
169.50 

167.243 

168.718 
170.142 
171-535 

444.616 
443.485 
442.393 
441.325 

8.00 

8.25 
8.50 

8.75 

6080 
6270 
6460 
6650 

82672.0 
85255.5 
87839.0 
90422.5 

170.81 
172.10 
173.35 

174.57 

172.888 
174.221 
175.514 
176.775 

440.389 
439.369 
438.380 
437.315 

9.00 
9.25 
9.50 
9.75 

6840 
7030 
7220 
7410 

93006.0 

95589.5 

98173.0 

100756.5 

175.77 
176.94 
178.08 
179.31 

178.017 
179.338 
180.408 
181.579 

436.366 
435.440 
434.539 
433.645 

10.00 
10.25 

10.50 
10.75 

7600 
7790 
7980 
8170 

103340.0 
105923.5 
108507.0 
111090.5 

180.31 
181.38 
183.44 
183.48 

183.719 
183.828 
184.927 
186.005 

433.775 

431.928 
431.090 
430.267 

11.00 
11.25 
11.50 
11.75 

8360 

8550 
8740 
8930 

113674.0 

116257.5 
118841.0 
121434.5 

184.50 
185.51 
186.49 

187.46 

187.065 
188.113 
189.131 
190.139 

439.460 
428.661 
427.886 
437.1,19 

12.00 
13.25 
12.50 
13.75 

9130 
9310 
9500 
9690 

124008.0 
126591.5 
129175.0 
181758.5 

188.41 
189.35 
190.27 
191.18 

191.126 
193.104 
193.060 
194.007 

436.368 
435.624 

424.896 
424.177 

13.00 
13.25 
13.50 
13.75 

9880 
10070 
10260 
10450 

134342.0 
136935.5 
139509.0 
143092.5 

192.08 
192.96 
193.83 
194.69 

194.944 
195.860  • 
196.766 
197.662 

423.465 
422.769 
422.080 
421.400 

14.00 

10640 

144676.0 

195.53 

198.537 

420.736 

THEBMOD  YNAMIC8. 


627 


TABLE  11.— continued. 

PRINCIPAL    TABLE    FOR    SATURATED    STEAM. 


7 

8 

. 

10 

11 

12 

Outer  latent 

heat 

A2JU, 

page  393. 

Valu 

M 

Cub.  meters  per 

kil.. 

page  391. 

esof 

u 
page  393. 

DifEerence. 

Kil  per  cub. 

meter, 

page  396. 

DifEerence. 

45.070 
45.163 
45.350 
45.337 

0.3643 
0.3556 
0.3475 
0.3400 

1683.0 
1735.3 

1787.1 
1838.7 

53.3 
51.9 
51.6 

3.7711 

3.8974 
4.0834 
4.1490 

0.1863 
1360 
1356 

45.430 
45.501 
45.578 
45.654 

0.3339 
0.3363 
0.3300 
0.3141 

1890.1 
1941.3 
1993.1 
3043.8 

51.4 
51.1 
50.9 
50.7 

4.3745 
4.3997 
4.5348 
4.6495 

1855 
1358 
1351 

1847 

45.737 
45.798 
45.868 
45.935 

0.3085 
0.3031 
0.1981 
0.1933 

3093.3 
3143.5 
3193.5 
3343.3 

50.5 
50.3 
50.0 
49.8 

4.7741 
4.8985 
5.0386 
5.1466 

1346 

1344 
1341 
1340 

46.001 
46.064 
46.137 
46.189 

0.1887 
0.1844 
0.1803 
0.1763 

3393.0 

3343.5 
3391.7 
3440.7 

49.7 
49.5 
49.8 
49.0 

5.3704 
5.3941 
5.5174 
5.6405 

1838 
1337 
1833 
1331 

46.347 
46.306 
46.363 
46.417 

0.1735 
0.1689 
0.1654 
0.1631 

3489.5 
3538.3 
3586.8 
3635.3 

48.8 
48.7 
48.6 
48.4 

5.7636 

5.8864 
6.0093 
6.1318 

1331 

1338 
1838 
1336 

46.471 
46.534 
46.576 
46.636 

0.1589 
0.1558 
0.1539 
0.1500 

3683.4 
3731.4 
3779.3 

3837.0 

48.3 
48.0 
47.9 

47.7 

6.3543 
6.3765 
6.4986 
6.6806 

1335 
1883 
1881 
1830 

46.676 
46.734 
46.773 
46.818 

0.1473 
0.1447 
0.1431 
0.1397 

3874.5 
3933.0 
3969.3 
3016.5 

47.5 
47.5 
47.3 

47.3 

6.7484 
6.8643 
6.9857 
7.1073 

1818 
1318 
1815 
1315 

46.864 

0.1373 

3063.4 

46.9 

7.3383 

1311 

628 


THERMOD  YNAMIC8. 


TABLE  lla. 

SATURATED    STEALI 


1 

2 

3 

4 

5 

1           6 

Tempera- 
ture 
C. 

t 

Absolute 
tempera- 
ture 
T 

Pressure  in 

millime- 
ters of    ba- 
rometer 
V 

Total  heat 

W 
(Regnault) 

Differ- 
ences. 

Heat  of 

liquid 

(J 

(Regnault) 

Differ- 
ences. 

Total  latent 

heat 
r=W-c( 

Differ- 
ences. 

0 

5 

10 

15 

20 

373 

378 
383 
388 
393 

4.600 

6.534 

9.165 

13.699 

17.391 

603.500 
608.025 
609,550 
611.075 
612.600 

1.535 

0.000 

5.000 

10.003 

15.005 

30.010 

5.000 
5.002 
5.003 
5.005 

608.500 
603.085 
599.548 
596.070 
592.590 

3.475 
3.477 
3.478 
3.480 

25 
30 
35 
40 
45 

398 
303 
308 
313 
318 

33.550 

31.548 
41.837 
54.908 
71.390 

614.125 
615.650 
617.175 
618.700 
620.225 

;j 

35.017 
30.026 
35.037 
40.051 

45.068 

5.007 
5.009 
5.011 
5.014 
5.017 

589.108 
585.684 
583.138 
578.649 
575.157 

3.483 
3.484 
3.486 
3.489 
3.493 

50 
55 
60 
65 

70 

333 
338 
333 
338 
343 

91.980 
117.475 
148.786 
186.938 
233.083 

621.750 
633.375 
634.800 
636.335 

637.850 

\\ 

50.037 
55.110 
60.137 
65.167 
70.201 

5.019 
5.033 
5.037 
5.030 
5.034 

571.663 
568.165 
564.663 
561.158 
557.649 

3.494 
3.498 
3.502 
3.505 
3.509 

75 
80 
85 
90 
95 

348 
353 

358 
383 
368 

288.500 
354.616 
433.003 
535.393 
633.693 

639.375 
630.900 
633.435 
633.950 
635.475 

;; 

75.239 
80.383 
85.389 
90.381 
95.438 

5.038 
5.043 
5.047 
5.053 
5.057 

554.136 

550.618 
547.096 
543.569 
540.037 

8.513 
3.518 
3.522 
3.537 
3.533 

100 
105 
110 
115 
120 

373 

378 
383 
388 
393 

760.000 

908.410 

1075.370 

136.9.410 

1491.380 

637.000 
638.535 
640.050 
641.575 
643.100 

;; 

100.500 
105.568 
110.641 
115.731 
130.808 

5.082 
5.068 
5.073 
5.080 

5.085 

536.500 
533.957 
539.409 
535.854 
533.394 

8.537 
3.543 
3.548 
3.555 
8.560 

135 
130 
135 
140    , 
145 

398 
403 
408 
413 
418 

1743.880 
2030.380 
3353.730 
3717.630 
3125.550 

644.635 
646.150 
647.675 
649.300 
650.725 

<< 

135.898 
130.997 
136.103 
141.315 
146.334 

5.093 
5.099  ! 
5.103 
5.113  ! 
5.119 

518.737 
515.153 
511.572 
507.985 
504.391 

8.567 
3.574 
3.581 
3.587 
3.594 

150 
155 
160 
165 
170 

423 

438 
433 
438 
443 

3581.230 
4088.530 
4651.620 
5274.540 
5961.660 

652.350 
658.775 
655  300 

656.825 
658.350 

';'; 

151.463 

156.598 
161.741 
166.893 
173.053 

5.138 
5.136 
5.143  i 
5.151 
5.160 

500.788 
497.177 
493.559 
489.933 
486.298 

3.603 
3.611 
3.618 
3.626 
3.035 

175 

180 
185 
190 
195 

448 
453 
458 
463 
468 

6717.430 
7546.390 
8453.230 
9443.7C0 
10519.630 

659.875 
661.400 
663.925 
664.450 
665.975 

I 

177.220 

183.398 
187.584 
193.780 
197.985 

5.168 
5.178 
5.186 
5.196 
5.805 

■  482.655 
479.003 
475.341 
471.670 

467  990 

3.643 
8.653 
3.661 
3.671 
3.680 

200 

473 

11688.960 

667.500 

" 

303.300 

5.315 

464.300 

3.690 

THEBMOD  TNAMIC8. 


629 


TABLE  Ila. — contiymed. 

SATURATED    STEAM. 


r 

8 

i 

9 

[ 

10 

1 

Outer  latent 

Differ- 

Steam heat 

Differ- 

Inner latent 

Differ- 

Yalne of 

Tem- 

lieat 

ence. 

J=  W- 

ence. 

heat 

ence. 

peia- 

A  p  u 

A  2>  11 

P  =  i'-Apii 

i 

u 

p_ 

■u 

C.  " 

i 

31.071 

575.43 

575.43 

210.66 

2.732 

0 

31.475 

0.404 

576.55 

1.12 

571.55 

3.88 

150.23 

3.805 

.      5 

31.893 

0.417 

577.66 

1.11 

567.66 

3.87 

108.51 

5.231 

10 

32.318 

0.426 

578.70 

1.10 

563.75 

3.91 

1    79.346 

7.104 

15 

32.755 

0.437 

579.84 

1.08 

559.83  ■ 

3.92 

58.720 

9.532 

20 

33.201 

0.44G 

580.93 

1.08 

555.91 

3.92 

43.963 

12.645 

25 

33.650 

0.455 

581.99 

1.07 

551.97 

3.94 

33.266 

16.592 

£0 

34.119 

0.463 

583.03 

1.07 

548.02 

3.95 

25.430 

21.545 

£5 

34.588 

0.469 

584.11 

1.05 

544.06 

3.96 

19.644 

27.696 

40 

35.064 

0.476 

585.16 

1.05 

540.09 

3.97 

15.315 

35.264 

45 

35.544 

0.480 

586.21 

1.05 

536.12 

3.97 

12.049 

44.492 

50 

36.027 

0.483 

587.25 

1.04 

532.14 

3.98 

9.5613 

55.646 

55 

3(i.512 

0.485 

588.29 

1.04 

528.15 

3.99 

7.6531 

69.020 

CO 

36.990 

0.484 

589.33 

1.04 

524.16 

3.99 

0.1711 

84.C38 

65 

37.478 

0.482 

590.37 

1.04 

520.17 

3.99 

5.0139 

103.746 

70 

37.955 

0.477 

591.42 

1.05 

516.18 

8.99 

4.1024 

125.825 

'io 

38.425 

0.470 

592.47 

1.05 

512.19 

3.99 

3.3789 

151.587 

80 

38.885 

0.460 

593,54 

1.07 

508.21 

3.98 

2.8003 

181.482 

85 

39.332 

0.447 

594.62 

1.08 

504.24 

3.97 

2.3344 

216.003 

SO 

39.762 

0.430 

595.71 

1.09 

500.27 

3.97 

1.9560 

255.687 

95 

40.200 

0.438 

596.79 

1.08 

486.29 

3.98 

1.6496 

800.856 

loo 

40.631 

0.431 

597.89 

1.10 

492.33 

3.96 

1.3978 

352.218 

105 

41.048 

0.417 

599.00 

1.11 

488.36 

3.97 

1.1903 

410.292 

110 

41.457 

0.409 

600.12 

1.12 

484.40 

3.96 

1.0184 

475.653 

115 

41.858 

0.401 

601.24 

1.12 

480.44 

3.96 

0.8752 

548.902 

120 

42.250 

0.392 

602.37 

1.13 

,  476.48 

3.96 

0.7555 

630.685 

125 

42.634 

0.384 

603.52 

1.15 

472.52 

3.96 

0.6548 

721.607 

180 

43.010 

0.376 

604.66 

1.14 

468.56 

3.96 

0.5698 

822.321 

135 

43.377 

0.367 

605.82 

1.16 

464.61 

8  95 

0.4977 

933.475 

140' 

43.735 

0.358 

606.99 

1.17 

460.60 

3.95 

0.4303 

1055.726 

145 

44.086 

0.351 

608.16 

1.17 

456.79 

3.96 

0.3839 

1189.736 

150 

44.428 

0.342 

609.35 

1.19 

452.75 

3.95 

0.3388 

1336.166 

155 

44.761 

0.333 

610.54 

1.21 

448.80 

3.95 

0.3001 

1495.686 

160 

45.086 

0.325 

611.74 

1.20 

444.85 

3.95 

0.2665 

1668.926 

165 

45.403 

0.317 

612.95 

1.21 

440.89 

3.96 

0.2375 

1856.538 

170 

45.711 

0.308 

614.16 

1.21 

436.94 

3.95 

0.2122 

2059.147 

175 

46.012 

0.299 

615.39 

1.23 

432.99 

8.95 

0.1901 

2277.366 

180 

46.304 

0.292 

616.62 

1.23 

429.04 

3.95 

0.1708 

2511.787 

185 

46,589 

0.285 

617.86 

1.24 

425.08 

3.96 

0.1538 

2762.974 

190 

46.8G4 

0.275 

619.11 

1.25 

421.13 

3.85 

0.1389 

3031.464 

195 

47.133 

0.269 

620.37 

i.26 

417.17 

3.96 

0.1257 

8817.795 

200 

630 


TEEBMOB  TNAMIC8. 


TABLE  III. 

AUXILIARY  TABLES   FOE    SATURATED   STEAM   (ZETJNER). 


Pressure 

P 

Q 

X 

r 

~if' 

r 

in  atmos- 
pheres. 

(Page  411) 

(Page  421) 

(Page  379) 

(Page  411) 

Apa 

0.5 

126.747 

0.26273 

549.404 

1.54887 

14.229 

1 

148.47 

0.31356 

536.500 

1.43834 

13.344 

2 

170.639 

0.36814 

521.866 

1.32588 

12.453 

4 

183.778 

0.40205 

512.353 

1.25913 

11.935 

4 

193.163 

0.42711 

505.110 

1.21129 

11.568 

5 

200.457 

0.44693 

499.186 

1.17395 

11.284 

6 

206.394 

0.46392 

494.124 

1.14322 

11.052 

7 

211.481 

0.47840 

489.686 

1.11714 

10.856 

8 

215.862 

0.49130 

485.709 

1.09441 

10.684 

9 

219.726 

0.50270 

482.093 

1.07425 

10.535 

10 

223.178 

0.51297 

478.776 

1.05817 

10.401 

11 

226.292 

0.52266 

475.707 

1.03980 

10.280 

13 

229.134 

0.53150 

472.839 

1.02477 

10.168 

13 

231.752 

0.53975 

470.141 

1.01088 

10.066 

14 

234.165 

0.54744 

467.600 

0.99801 

9.971 

THEBMOD  YNAMIC8. 


631 


Eh        h 


Pressure 

above   a 

vacuum,    iu 

per 
square 
iueh. 

::!2S;5 

a 

> 

ipmiiii 

1 

3 

! 

i 

m 

liiiiiiliii 

i 

1 

i 

3 

o 

|I|I4 

03        c 

ii^'i 

i 

1 

1  11 

llliiiiiii 

iiii 

1:3 

C5  jC5 

llii 

S9|gS|||oS 

IHI 

i^ 

III'' 

iiiliiliie 

iiii 
llii 

11 

] 

i 

i 

2"s 

S  o  m 

iir 

i 

i 

rH(Mm^intot-coc;o 

SS2^ 

s. 

632 


THEBMOD  YNAMIC8. 


(5  " 

afll^' 

... 

s?3?5?;?/55S?.r5g 

SS??J?^55 

i 

> 

ifil 

11  III! 

iillS 

eotooccooc'oot^oo.-; 

xccc^oc-.oo 

C3  -^  'S  .2  «■ 

iiKSSlS^l 

HII3!-! 

Jiil 

§S|2| 

g|SS||2||| 

iiilil 

1 

s 

i 

pi 

CO      c 

iiiii 

cq  m  -*■  lO  Lo  ic  o  t-:  t^  00 

...... 

1  1 

IIIII 

iiiiii 

ilii 

«  00  ■=  m-o 

c;  «  ^  ^j  ^  ^  ^  «j  w  o 

o  lo  lo  id  si  ai 

lll^ 

iittiiiiit 

iilMl 

iiilsl 

III. 

iigiiiiiii 

iiiiiiiiii 

P 

1  ^   s^- 

i-2|R|l 

sssss 

SS3?5c5^S?5S§i 

Sg??:?i§S§ 

THERMO  D  YNAMIC8. 


633 


(CieoT*iic50f-xoiO 


<  (N  M  M<  m  CO  J- 


- 1-  i>  t^  t-  i.~ 


;  m  o  t-  o  c-_  C!  o  uo  I  c:  c-  oc  -^i  c;_  m  c-5 1--  lo  »o 

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!■-;  (>J  00  «0  to  00  CJ  00  m  M 


W^OOOtH  00 
-^  05  "^  O  lO  tH  O 


:  ^  ^  ??  ^  ^  ?"1  ©* 


>  c^  Ci  L—  io  CO '; 


Oi-HOSOlOWMTt'O^O 

ooq!>-^_in-*ro(N.-o 
«:cDt6cDo:0!:otoccc6 


d.  r^^g5omjot-o 


)  (M  CO  <M  ^  -*  C 


i  ^  Ai  ^  ^  ,ik  =^  s,i  -„  I  3  ^  ^  ^  2  r,  2  *-  "  "t    rf «  c  i-  5?  lo  ^ 

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o  00  <o  « 
o  HHcn- 
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xmciOiopp'-i     T-i 


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d5o5o;ooioiooi-ic 
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»<Mi-IIM  00  ■ 


-_5  <>!  OJ  —  O  S  00 


cc  t'  :o  ic  lo  -^  CO  CO  <^*  T-<* 

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in  Ci  o  00  CO  in  rti 

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d  d ; 


)  low  m  t- t 


30  Ol 
c-^t-^OOOOOJCCCOOO 


'^coooffioit-coooNin 

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T);incoi>.£-ooo;cnorH 

ocodooodooccpcccdo 


o:c:  c;  pose: 


Tit— tClCOt-t-COCOCCC 


)  00  S  00  c 


c^C3coinOT-iooc<i-*i 
crco^ivaoco,-HiocoL 

C-.  CO  00  Ol  CO  O  ^  t-;  C  C 

t-'  c^  d  ci  CO  lo  d  c^  d  c 
c"iCTcM<Mc<<dcqiri(Nc 


■CD  OS 

c5  oici 


cortCOint-wCTOO  oocococow^eo^CT-J 
c^<^^'-^a>mocOTl^  ^i^coocoT^^cot-coi 
rtCOincoooo^CTl  co^miococococcccc 


SfeS? 


§g 


j-rairico-^oco-^coo 
ii-iiocncot-ococoos 
idT-lcq-^uoc-^oodd 

;t-t-Ir-t-t-t-t~C-00 
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THi-ICJlT-llOrHTH-^COin 
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rH  c^  in  c-  o:  I-l<^^  CO  •»  in 
(?i  CO*  'ti*  in  d  00  d  d  T-H  nj 
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1MSQ1M(M<M(NIM5J(N« 


t^  CO  CO  in  rr  GO  ni 
CO  ^  in  d  t-  00  cr; 

CO  CO  CO  CO  CO  50  CO 


wci«'*incot-c»05^    ^-^'^>coT^mco^-GOc:o    r-'f^jcc^incot- 
intoinSinSihinin»    cocDcoococo^cocoi-    t-i>i>£>i-i>t- 


634: 


THEBMOD  TNAMIGS. 


ii 

vacuum,    m 
pounds 

per 
square 
inch. 

ggg 

SSSSSgfeggS 

SggSggfeggS 

3 
> 

S° 

equal 

weight  of 
distilled 
water  at 
tempera- 
ture of 

maximum 
density. 

342  6 
338.5 
334.5 
330.6 
336.8 
333.1 
319.5 
315.9 
312.5 
309.1 
305.8 
302.5 
299.4 

296.3 
293.2 
290.3 
287.3 
284.5 
281.7 
279.0 
276.3 
273.7 
271.1 

Of  a 
pound  of 
steam  in 

cubic 

feet. 

5.488  . 

5.433 

5-358 

5.396  ' 

5  335 

.5.176 

5.118 

5.061 

5.006 

4.951 

4.898 

4.846 

4.796 

4.746 
4.697 
4.6.50 
4.603 
4.5.57 
4.513 
4.469 
4.426 
4.384 
4.343 

■Weight  of  a 

cubic  foot    of 

steam  in 

pounds. 

lis 

.188-33 
.191017 
.193310 
.19.5401 
.197.591 
.1997'8t 
.301969 
.3011.55 
.306340 
.308.535 

.3107'09 
.313,893 
.21.5074 
.217-2.53 
.319130 
.231(;04 
.33377-8 
.235950 
338123 
.330393 

i 

o 

i 

< 

1 
1 

M 

mil 

o        II 
H        5, 

III 

894.491 
893.879 
893.375 

893;08.3 
891.496 
890.913 
890.335 
889.763 
,889.196 

888.633 
8,88.075 
f-87..531 
8,86.973 
886.437 
8  5  887 
885.352 
884  831 
881.395 
883.773 

llii 

79.583 
79.639 
79.695 
79,749 
79.S03 
79.8.56 
79.909 
79961 
80.013 
80.063 
80.113 
80  1(;3 
80.310 

8N.358 
80.305 
80.351 
,80.397 
811.443 
80.4,87 
80.531 
80..57'6 
81).  630 
80.665 

III 

814.742 
814.07-7 
813.419 
813.768 
813.133 
811.484 
810.850 
810.223 
8(19.601 
808.986 

808.375 
807.770 
807.170 
806.575 
805. 9S5 
805.400 
804., K31 
804.315 
803.675 
803  108 

Keqniredto 
raise  the  tem- 
perature of  the 
water  from  32° 
to  f^ 
1- 

282.8.30 
283.701 
284.563 
285.414 
286.260 
287.096 
287.937 
288.7.50 
289.565 
290.373 

291.176 
291.970 
293.758 
298..539 
294.314 
295.083 
295.845 
296.601 
297.3,50 
298.093 

2.- 

0.123 
1.000 
1.866 
3.725 
3.576 
4.417 
5.250 
6.076 
6.893 
7.705 
8..510 
9.306 
20.094 

330.877 
321.6,53 
322.422 
323.183 

324'.688 
325.431 
326.169 
326.900 
327.625 

CO  COCO  cococococococococcco 

1 

11 

vacuum 

in 
pounds 

per 
square 
inch. 

sr-8 

x3?S3SSfegBSS 

SSSS^gfegsS 

THEBMOD  TNAMICS. 


635 


ll^^MT^<lno^.ooc50 


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300330 


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INDEX. 


Absolute  zero,  142. 

Adiabatic  compression  of  nir,  167. 

curve,  approximate  formula  for,  433. 
for  air,  1.5S. 
for  steam,  421. 
for  superheated  steam,  536. 
efllux  of  steam,  473. 
expansion  of  air,  165. 

of  steam,  430. 
transfer  of  air,  172. 
Air,  compressed,  322. 
efflux  of,  313. 
engines,  compressed,  322. 

complete  expansion, 

328. 
full  pressure,  334. 
incomplete      expan- 
sion, 338. 
friction  of.  in  pipes,  327. 
Atmospheres  into  pounds  and  kilograms,  355. 
Atmospheric  gas  engine,  294. 
Back  pressure,  work  of,  579. 
Bechai-d,  experiments  of,  48. 
Bernoulli,  Daniel,  52. 
Body  tension,  122. 
Boiler,  locomotive.  4.53. 

steam,  generation  of  steam  in,  451. 
Boiling  point,  143. 
Bnflon,  hvpotliesis  of,  50. 
Caloric,  105. 
Capacit}',  volume,  130. 
Carnot,  53. 
Charles,  law  of,  63. 

Chlorophj'll,  absorption  spectrum  of,  87. 
Clapeyron,  54. 

Clausing,  views  as  to  nature  of  heat,  113. 
Coefficient  of  irregularitv,  590. 
Colding,  54. 
Compressed  air  engines,  322. 

complete     expansion, 

328. 
full  pressure.  334. 
incomplete  expansion, 
338. 
Compression  of  gasc^.  14f>. 

of  steam,  adiabatic,  4-30. 
Compressors,  air,  .322. 
Condensation  of  steam  in  expanding,  72. 
Condenser,  449. 

dimensions  of,  601. 
jet,  465. 
surface,  460. 
theory  of.  460. 
Conduction  and  radiation  of  heat.  115. 
Connecting  rod,  influence  of  length  of,  t)92. 
Constant  steam  weight,  curve  of,  399. 


[  Constant  volume,  addition  of  heat  under,  448. 
Contraction  of  bodies  when  heated,  66. 
Convection,  electrolytic,  83. 
Coriolis.  theorem  of,  63. 
1  Crank,  theory  of,  587. 
Critical  temperature,  401. 
Cost  of  working  of  steam  engine,  604. 
Cycle  process,  19. 

of    steam   engine,  imperfection 

of,  555. 
of  the  steam  engine,  550. 
simple,  reversible,  179. 
C.vlinder,  action  of  steam  in,  565. 
Davy,  experiments  of,  53, 102. 

views  as  to  nature  of  heat,  113. 
Delivery  indicated,  584. 

useful,  599. 
Density  of  saturated  steam,  396. 
Disgregation  work,  18-25, 132. 

in  crystals  and  liquids,  64. 
Dynamide,  112. 
Effect,  mechanical,  3. 
Efficiency,  coefficient  of,  35. 

of  steam  engine,  206. 
Efflux,  adiabatic,  of  steam,  473. 
of  air  from  vessels,  313. 
of  hot  water,  481. 
of  steam,  470. 

velocity  of,  477. 
Electrolytic  convection,  83. 
Electro-magnetic  engine,  39. 

forces,  nature  of,  80. 
Engine,  electro-magnetic,  39. 
Ericsson's,  73. 
hot-air,  32. 
Engines,  hot-air,  75. 
steam,  546. 
Engine,  work  of,when  disconnected,  587. 
English  measures  inio  French,  353. 
Entropy,  92, 187. 
Ericsson's  engine,  73. 

hot-air  engine,  213. 
Ether,  111. 

Evaporation,  action  of  heat  in,  3.59. 
work  of  water  in,  207. 
Examples  for  practice,  346. 
Expansion,  coefficient  of,  140. 
degree  of,  57.5. 

of  air  under  constant  pressure,  175. 
of  Erases,  137. 
of  steam,  402. 

adiabatic,  4.30. 
Favre,  experiments  of,  10. 
Fill,  coefficient  of,  604. 
Fizeau  and  Foucault,  experiments  of,  7. 
Fly-wheel,  rim  and  arms,  594. 

639 


640 


INDEX. 


Fly-wheel,  weiirht  of.  589. 
Foot-pounds  into  meter-kilograms,  354. 
Formute,  recapitulation  of,  203. 
French  measures  into  English,  353. 
Friction,  6. 

of  air  in  pipes,  327. 

of     blood,    influence     upon    animal 

heat,  85. 
pressure  for  overcoming,  596. 
Fuel  used  per  hour,  602. 
Gas  engine,  atmospheric.  294. 

of  Otto  and  Sanger,  280. 
Gases,  compression  of,  146. 
constitution  of,  (>7. 
expansion  of,  137. 
law  of  expansion  of,  63. 
speciflc  heat  of,  146. 
Gay-Lussac's  law.  147. 
Gravity,  specific.  130. 
Heat,  action  of,  in  evaporation,  359. 
actual  speciflc,  of  water,  376. 
addition  under  constant  volume,  448. 
a  kind  of  motion'.  111. 
calculatioa  of  mechanical  equivalent  of, 

143. 
conduction  and  radiation,  115. 
different  works  performed  by.  121. 
fundamental  equations  of  mechanical  the- 
ory of,  123. 
generated  by  mechanical  action,  103. 
identity  with  light,  7.  ' 

inner  and  outer,  of  vaporization,  379. 
latent,  18. 

law  of  transmission  of,  76. 
mean  speciflc,  of  water,  375. 
mechanical  equivalent  of,   10,  44,  65,  70, 

80.  104.  106. 
of  friction,  101. 
of  liquids,  ,373. 
of  vaporization,  .377. 
outer  and  inner  latent,  of  steam,  392. 
rays,  interference  of,  7. 
specific,  138, 134. 

of  gases,  146. 
of  water,  374. 
steam,  383. 
total,  377. 

transformation  into  work,  79. 
unit,  105. 

views  as  to  the  nature  of,  113. 
weight.  187,  441. 
Heating  surface.  452. 
Him,  experiments  of,  61, 103. 
Hirn's  experiments,  13. 

law  for  superheated  steam,  508. 
Horse  power,  cost  of,  603. 

Hot  air  and  steam,  comparison  of  work  of,  210. 
engines  comparison  of,  206. 
engine,  32,  75. 

Ericsson's,  213. 

formulae  for,  251. 

historical,  209. 

maximum  delivery,  341,  248. 

of  Laubereau  and  Lehmann,  256. 

of  Lehmann,  267. 

of  Sterling,  343. 

of  linger,  254. 

open,  with  open  fireplace,  213. 

interior  fire,  233. 
regenerator,  73. 
theory  of,  228. 
Hugon's  engine,  293. 
Inches  into  centimeters,  853. 
Indicated  delivery,  .584. 
Indicator,  stt-ani,  13. 
Induction  phenomena,  77. 
Injector,  description  of,  490. 

theory  of,  491. 
Inner  and  outer  lieat  of  vaporization,  379. 
work,  131,  122. 
work,  graphical  representation  of,  199. 


'  Intermediate  body  in  cycle  process,  181. 
In'egularity,  coeflicient  of,  590. 
Isenergic  curve  for  air,  1.57. 
Isenrropic    "         "        158. 
Isodynamic  "         "        157. 

steam,  418. 

superheated  steam,  538. 
Isonietric  curve,  176. 
Isopiestic      "       176. 
Isothermal    "      for  air,  154. 

for  steam,  415. 
for  superheated  steam,  539. 
Jet  condenser,  465. 
Joule  and  Pavre.  experiments  of,  40. 

experiments  of,  9,  71,  104. 
Journals,  diameter  of,  595. 
Kilograms  into  pounds,  353. 

per   square   centimeter  into  pounds 
per  square  inch,  355. 
Latent  heat,  18. 

outer  and  inner  of  steam,  393. 
Lauliereau  and  Lehmann,  hot-air  engine,  356. 
Laubereau's  engine,  delivery  of,  264. 
I  dimensions  of,  266. 

i  theory  of,  360. 

Lavoisier  and  Laplace,  .53. 
Lehmann's  engine,  delivery  of,  274. 

hoi  -air  engine,  267. 
Length  of  connecting  rod,  influeucs  of,  592. 
Lenoir  engine,  delivery  of,  386. 
Light,  identity  witli  heat,  7. 
Liquid,  heat  of,  373. 
Locomotive  boiler,  453. 
Magnus,  formulaj  of,  369. 
Mafiotte  and  Gay-Lussac's  laws  combined,  148, 

1-)1. 
Mariotte's  law,  146. 
Mayer,  vie^Vs  of,  89, 103. 
Mazeline,  hot-air  engine  of,  223. 
Mechanical  effect,  3. 

equivalent  of  heat,  10,  44,  65,  70,  8D, 
104,  106. 

calculation       o.'', 
148. 
theory  of  heat,  fundamental  equa- 
tions, 133. 
Melting  point,  143. 
Meter-kilograms,  into  foot-lbs.,  354. 
Meters  into  inches,  353. 
Mixture  of  steahi,  455. 
Motion,  perpetual,  5,  59. 
Notation,  customary  for  steam,  386. 

of  frequent  use,  202. 
Otto  and  Langen,  gas  engine,  280. 
Outer  and  inner  work,  121, 122. 

work,  18. 
Passages,  steam,  cross-section  of,  583. 
Perpetual  motion,  5,  59. 
Piston,  mean  velocity  of,  .586. 
Pounds  into  kilograms,  3.54. 

per  square   inch   into   kilograms   per 
square  centimeter,  355. 
Pressure,  back,  work  of,  579. 

change  of,  with  volume  for  air,  194. 
speciflc,  122. 
Process,  cycle,  19. 
Pumps,  dimensions  of,  601. 
R  idiatifin  and  conduction  ol'heat,  115. 
Redtenbacher,  theory  of.  111. 
Reduction  tables,  353. 
Regenerator,  247. 

in  hot-air  engines,  73. 
Regnault,  experiments  of,  367. 

fonnuliBof,  368. 
KOntgen's  "  370. 

Rumf ord,  experiments  of,  53, 101. 
Saturated  steam,  364. 

density  of,  396. 
formulse  for,  366. 
Saturation,  curve  of,  401. 
Specific  gravity,  130. 


INDEX. 


641 


Specific  heat,  138, 134. 

actual,  of  water.  376. 
mean,  of      "       375. 
of  gases,  146. 
of  water,  374. 
pressure,  183. 
steam  volume,  385. 

calculated,  388. 
volume,  132. 

of  superheated  steam,  509. 
Steam,  adiabatic  compression,  of  430. 
curve  for,  431. 
expansion  of,  430. 
and  hot  air,  comparison  of  work  of,  210. 
condensation  of,  in  expanding,  72. 
efflux  of,  470. 
engine,  546. 

and  hot-air  engine,  comparison 

of,  206. 
calculation  of,  605. 
complete  calculation  of,  565. 
cost  of  working,  604. 
cycle  process  of,  550. 
efficiency  of,  206. 
imperfection  of   cycle   process 

of,  555. 
motive  power  of,  60. 
perfect,  550. 
expansion  of,  402. 
gas,  365. 

feneral  properties  of,  363. 
eat,  383. 
indicator,  13. 

isodynamic  cun'e  for,  418. 
isothermal  curve  for,  415. 
mixtures  of,  455. 


superheated,  365, 506. 

adiabatic  curve  for,  536. 
isodynamic  curve  for,  5.38. 
isothermal  curve  for,  539. 
volume,  calculated,  388. 
per  stroke,  575. 
specific,  385. 
weight,  curve  of  constant,  399. 

per  hour,  600. 
work  of  the  driving,  577. 
41 


Sterling's  engine,  243. 
Superheated  steam,  365, 506. 

adiabatic  curve  for,  536. 
isodynamic  curve  for,  £ 
isothennal  curve  for,  f 
specific  volume,  509. 
Zeuner's  theory  of,  517. 
Surface  condenser, 460. 

healing,  452. 
Tables,  reduction,  362. 
Temperature,  absolute  zero  of,  142. 

critical,  401. 
Tension,  body,  123. 
Thermodynamic  function,  187. 
Thermodynamics,  definition  of,  3. 
Thomson  and  Joule,  experiments  of,  71. 

Wm.,  50. 
Total  heat,  377. 
Unger's  hot-air  engine,  254. 
Useful  delivery,  599. 
Vaporization,  heat  of,  377. 

inner  and  outer  heat  of,  379. 
Vegetation,  dependence  upon  light,  85. 
Velocity  of  eflflux  of  steam,  477. 

of  piston,  mean,  586. 
Vis  viva,  3. 
Volimae  capacity,  130. 

change  of,  with  pressure,  194. 
specific,  122. 

of  superheated  steam,  509. 
steam.  385. 
calculated,  388. 
steam,  per  stroke,  575. 
Water,  hot,  efflux  of,  481. 
Work,  3. 

disgi-egation,  18,  25, 132. 

in  crystals  and  liquids, 
inner  and  outer,  121. 
outer,  18. 

and  inner,  122. 
performed  by  heat,  106. 
uselul,  6. 
Working  of  steam  engine,  cost  of.  604. 
Young,  53. 
Zero,  absolute,  142. 

Zeuner,  theory  of  snperhcated  steam,  517. 
Zinc,  decomposition  of,  84. 


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BOSTON  COLLEGE 

•^HrSICS  DEPT. 


2G0217. 


Date  Due 


BOSTON  COLLEGE 


3  9031  01482072  4 


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